A T-SQL stored procedure for moving an index from one file group to another. The original script was found at http://blogs.msdn.com/b/ramoji/archive/2008/09/26/how-to-move-existing-indexes-from-one-filegroup-to-another.aspx and updated according to responses found on the same page.

gistfile1.sql

IF  EXISTS (SELECT * FROM sys.objects WHERE object_id = OBJECT_ID(N'[dbo].[MoveIndexToFileGroup]') AND type in (N'P', N'PC'))
	BEGIN
		DROP PROCEDURE [dbo].[MoveIndexToFileGroup]
	END
GO

CREATE PROC [dbo].[MoveIndexToFileGroup] (        
	@DBName sysname,   
	@SchemaName sysname = 'dbo',       
	@ObjectNameList Varchar(Max),        
	@IndexName sysname = null,  
	@FileGroupName varchar(100)  
) WITH RECOMPILE

AS        
  
BEGIN  
     
	SET NOCOUNT ON;
  
	DECLARE @IndexSQL NVarchar(Max)  
	DECLARE @IndexKeySQL NVarchar(Max)  
	DECLARE @IncludeColSQL NVarchar(Max)  
	DECLARE @FinalSQL NVarchar(Max)  
  
	DECLARE @CurLoopCount Int  
	DECLARE @MaxLoopCount Int  
	DECLARE @StartPos Int  
	DECLARE @EndPos Int  
  
	DECLARE @ObjectName sysname  
	DECLARE @IndName sysname  
	DECLARE @IsUnique Varchar(10)  
	DECLARE @Type Varchar(25)  
	DECLARE @IsPadded Varchar(5)  
	DECLARE @IgnoreDupKey Varchar(5) 
	DECLARE @AllowRowLocks Varchar(5)  
	DECLARE @AllowPageLocks Varchar(5) 
	DECLARE @FillFactor Int  
	DECLARE @ExistingFGName Varchar(Max) 
	DECLARE @FilterDef NVarchar(Max)
  
	DECLARE @ErrorMessage NVARCHAR(4000)  
	DECLARE @SQL nvarchar(4000)  
	DECLARE @RetVal Bit  
  
	DECLARE @ObjectList Table(Id Int Identity(1,1),ObjectName sysname)  
  
	DECLARE @WholeIndexData TABLE (
		ObjectName SYSNAME
		,IndexName SYSNAME
		,Is_Unique BIT
		,Type_Desc VARCHAR(25)
		,Is_Padded BIT
		,[Ignore_Dup_Key] BIT
		,[Allow_Row_Locks] BIT
		,[Allow_Page_Locks] BIT
		,Fill_Factor INT
		,Is_Descending_Key BIT
		,ColumnName SYSNAME
		,Is_Included_Column BIT
		,FileGroupName VARCHAR(MAX)
		,Has_Filter BIT
		,Filter_Definition NVARCHAR(MAX)
		,key_ordinal TINYINT
	)

	DECLARE @DistinctIndexData TABLE (
		Id INT IDENTITY(1, 1)
		,ObjectName SYSNAME
		,IndexName SYSNAME
		,Is_Unique BIT
		,Type_Desc VARCHAR(25)
		,Is_Padded BIT
		,[Ignore_Dup_Key] BIT
		,[Allow_Row_Locks] BIT
		,[Allow_Page_Locks] BIT
		,Fill_Factor INT
		,FileGroupName VARCHAR(Max)
		,Has_Filter BIT
		,Filter_Definition NVARCHAR(Max)
	)
  
-------------Validate arguments----------------------   
  
	IF(@DBName IS NULL)  
		BEGIN  
			SELECT @ErrorMessage = 'Database Name must be supplied.'   
			GOTO ABEND  
		END  
  
	IF(@ObjectNameList IS NULL)  
		BEGIN  
			SELECT @ErrorMessage = 'Table or View Name(s) must be supplied.'   
			GOTO ABEND  
		END  
  
	IF(@FileGroupName IS NULL)  
		BEGIN  
			SELECT @ErrorMessage = 'FileGroup Name must be supplied.'   
			GOTO ABEND  
		END  
  
	--Check for the existence of the Database  
	IF NOT EXISTS(SELECT Name FROM sys.databases where Name = @DBName) 
		BEGIN 
			SET @ErrorMessage = 'The specified Database does not exist' 
			GOTO ABEND
		END
  
	--Check for the existence of the Schema  
	IF (upper(@SchemaName) <> 'DBO')
		BEGIN
			SET @SQL = 'SELECT @RetVal = COUNT(*) FROM ' + QUOTENAME(@DBName) + '.sys.schemas WHERE name = ''' + @SchemaName + ''''

			BEGIN TRY
				EXEC sp_executesql @SQL, N'@RetVal Bit OUTPUT', @RetVal OUTPUT
			END TRY
			BEGIN CATCH
				SELECT @ErrorMessage = ERROR_MESSAGE()
				GOTO ABEND
			END CATCH

			IF (@RetVal = 0)
				BEGIN
					SELECT @ErrorMessage = 'No Schema with the name ' + @SchemaName + ' exists in the Database ' + @DBName
					GOTO ABEND
				END
		END 
  
	--CHECK FOR THE EXISTENCE OF THE FILEGROUP  
	SET @SQL = 'SELECT @RetVal=COUNT(*) FROM ' + QUOTENAME(@DBName) + '.sys.filegroups WHERE name = ''' + @FileGroupName + ''''  
	BEGIN TRY  
		EXEC sp_executesql @SQL,N'@RetVal Bit OUTPUT',@RetVal OUTPUT  
	END TRY  
	BEGIN CATCH  
		SELECT @ErrorMessage = ERROR_MESSAGE()   
		GOTO ABEND  
	END CATCH  
  
	IF(@RetVal = 0)  
		BEGIN  
			SELECT @ErrorMessage = 'No FileGroup with the name ' + @FileGroupName + ' exists in the Database ' + @DBName   
			GOTO ABEND  
		END  
  
----------Get the objects from the concatenated list----------------------------------------------------  
  
SET @StartPos = 0  
SET @EndPos = 0  
  
WHILE(@EndPos >= 0)  
BEGIN  
  
 SELECT @EndPos = CHARINDEX(',',@ObjectNameList,@StartPos)  
 IF(@EndPos = 0) --Means, separator is not found  
 BEGIN  
  INSERT INTO @ObjectList  
  SELECT SUBSTRING(@ObjectNameList,@StartPos,(LEN(@ObjectNameList) - @StartPos)+1)   
     
  BREAK  
 END  
   
 INSERT INTO @ObjectList  
 SELECT SUBSTRING(@ObjectNameList,@StartPos,(@EndPos - @StartPos))  
    
 SET @StartPos = @EndPos + 1  
   
END  
  
-------------Check for the validity of all the Objects----------------------  
  
SET @StartPos = 1  
SELECT @EndPos = COUNT(*) FROM @ObjectList  
  
WHILE(@StartPos <= @EndPos)  
BEGIN  
  
 SELECT @ObjectName = ObjectName FROM @ObjectList WHERE Id = @StartPos  
  
 --CHECK FOR EXISTENCE OF THE OBJECT  
 SET @SQL = 'SELECT @RetVal=COUNT(*) FROM ' + QUOTENAME(@DBName) + '.sys.Objects WHERE type IN (''U'',''V'') AND name = ''' + @ObjectName + ''''  
 BEGIN TRY  
  EXEC sp_executesql @SQL,N'@RetVal Int OUTPUT',@RetVal OUTPUT  
 END TRY  
 BEGIN CATCH  
  SELECT @ErrorMessage = ERROR_MESSAGE()   
  GOTO ABEND  
 END CATCH  
  
 IF(@RetVal = 0)  
 BEGIN  
  SELECT @ErrorMessage = 'No Table or View with the name ' + @ObjectName + ' exists in the Database ' + @DBName   
  GOTO ABEND  
 END   
  
 --Check for existence of Index  
 IF(@IndexName IS NOT NULL)  
 BEGIN  
  SET @SQL = 'SELECT @RetVal=COUNT(*) FROM ' + QUOTENAME(@DBName) + '.sys.Indexes si INNER JOIN ' + QUOTENAME(@DBName) + '.sys.Objects so '  
  SET @SQL = @SQL + ' ON si.Object_Id = so.Object_Id WHERE so.Schema_id = ' + CAST(Schema_Id(@Schemaname) as varchar(25))   
  SET @SQL = @SQL + ' AND so.name = ''' + @ObjectName + ''' AND si.name = ''' + @IndexName + ''''   
  
  BEGIN TRY  
   EXEC sp_executesql @SQL,N'@RetVal Int OUTPUT',@RetVal OUTPUT  
  END TRY  
  BEGIN CATCH  
   SELECT @ErrorMessage = ERROR_MESSAGE()   
   GOTO ABEND  
  END CATCH  
  
  IF(@RetVal = 0)  
  BEGIN  
   SELECT @ErrorMessage = 'No Index with the name ' + @IndexName + ' exists on the Object ' + @ObjectName   
   GOTO ABEND  
  END  
 END  
   
 SET @StartPos = @StartPos + 1  
END  
  
-------------Loop till all the Objects are processed----------------------  
  
SET @StartPos = 1  
SELECT @EndPos = COUNT(*) FROM @ObjectList  
  
WHILE(@StartPos <= @EndPos)  
BEGIN  
  
 SELECT @ObjectName = ObjectName FROM @ObjectList WHERE Id = @StartPos  
 
 -------------Build the SQL to get the index data based on the inputs provided----------------------   
  


 SET @IndexSQL =   
 'SELECT so.Name as ObjectName, si.Name as IndexName,si.Is_Unique,si.Type_Desc'  
 + ',si.Is_Padded,si.Ignore_Dup_Key,si.Allow_Row_Locks,si.Allow_Page_Locks,si.Fill_Factor,sic.Is_Descending_Key'  
 + ',sc.Name as ColumnName,sic.Is_Included_Column,sf.Name as FileGroupName,'+ CASE WHEN @@VERSION LIKE '%Server 2005%' THEN '0 as Has_Filter, N'''' as Filter_Definition' ELSE 'si.Has_Filter,si.Filter_Definition' END +',sic.Key_Ordinal FROM '
 + QUOTENAME(@DBName) + '.sys.Objects so INNER JOIN ' + QUOTENAME(@DBName) + '.sys.Indexes si ON so.Object_Id = si.Object_id INNER JOIN '  
 + QUOTENAME(@DBName) + '.sys.FileGroups sf ON sf.Data_Space_Id = si.Data_Space_Id INNER JOIN '   
 + QUOTENAME(@DBName) + '.sys.Index_columns sic ON si.Object_Id = sic.Object_Id AND si.Index_id = sic.Index_id INNER JOIN '  
 + QUOTENAME(@DBName) + '.sys.Columns sc ON sic.Column_Id = sc.Column_Id and sc.Object_Id = sic.Object_Id '  
 + ' WHERE so.Name = ''' + @ObjectName  + ''''  
 + ' AND so.Schema_id = ' + CAST(Schema_Id(@Schemaname) as varchar(25)) + ' AND si.Type_Desc = ''NONCLUSTERED'' '  
  
 IF(@IndexName IS NOT NULL)  
 BEGIN  
  SET @IndexSQL = @IndexSQL + ' AND si.Name = ''' + @IndexName + ''''  
 END  
  
 SET @IndexSQL = @IndexSQL + ' ORDER BY ObjectName, IndexName, sic.Key_Ordinal'  
  
 --PRINT @IndexSQL  
  
 -------------INSERT THE INDEX DATA INTO A VARIABLE----------------------   
  
	BEGIN TRY  
		INSERT INTO @WholeIndexData
		EXEC sp_executesql @IndexSQL
	END TRY  
	BEGIN CATCH  
		SELECT @ErrorMessage = ERROR_MESSAGE()   
		GOTO ABEND  
	END CATCH  
  
 --Check if any indexes are there on the object. Otherwise exit  
 IF (SELECT COUNT(*) FROM @WholeIndexData) = 0  
 BEGIN  
  SELECT 'Object does not have any nonclustered indexes to move'   
  GOTO FINAL   
 END  
    
 -------------Get the distinct index rows in to a variable----------------------   
  
INSERT INTO @DistinctIndexData
SELECT DISTINCT  
	 ObjectName
	,IndexName
	,Is_Unique
	,Type_Desc
	,Is_Padded
	,[Ignore_Dup_Key]
	,[Allow_Row_Locks]
	,[Allow_Page_Locks]
	,Fill_Factor
	,FileGroupName
	,Has_Filter
	,Filter_Definition
FROM @WholeIndexData
WHERE ObjectName = @ObjectName;
  
 SELECT @CurLoopCount = Min(Id), @MaxLoopCount = Max(Id) FROM @DistinctIndexData WHERE ObjectName = @ObjectName
  
 --SELECT @CurLoopCount, @MaxLoopCount  
  
 -------------Loop till all the indexes are processed----------------------   
  
 WHILE(@CurLoopCount <= @MaxLoopCount)  
 BEGIN  
  
  SET @IndexKeySQL = ''  
  SET @IncludeColSQL = ''  
  
  -------------Get the current index row to be processed----------------------  
  SELECT   
   @IndName   = IndexName  
   ,@Type   = Type_Desc
   ,@ExistingFGName = FileGroupName
   ,@IsUnique  = CASE WHEN Is_Unique = 1 THEN 'UNIQUE ' ELSE '' END  
   ,@IsPadded  = CASE WHEN Is_Padded = 0 THEN 'OFF,' ELSE 'ON,'  END  
   ,@IgnoreDupKey = CASE WHEN Ignore_Dup_Key = 0 THEN 'OFF,' ELSE 'ON,' END  
   ,@AllowRowLocks = CASE WHEN Allow_Row_Locks = 0 THEN 'OFF,' ELSE 'ON,' END 
   ,@AllowPageLocks = CASE WHEN Allow_Page_Locks = 0 THEN 'OFF,' ELSE 'ON,' END  
   ,@FillFactor  = CASE WHEN Fill_Factor = 0 THEN 100 ELSE Fill_Factor END  
   ,@FilterDef  = CASE WHEN Has_Filter = 1 THEN (' WHERE ' + Filter_Definition) ELSE '' END  
  FROM @DistinctIndexData   
  WHERE Id = @CurLoopCount  
    
  -------------Check if the index is already not part of that FileGroup----------------------  
  
  IF(@ExistingFGName = @FileGroupName)  
  BEGIN  
   PRINT 'Index ' +  @IndName + ' is NOT moved as it is already part of the FileGroup ' + @FileGroupName + '.'  
   SET @CurLoopCount = @CurLoopCount + 1  
   CONTINUE  
  END  
  
  ------- Construct the Index key string along with the direction--------------------  
	SELECT @IndexKeySQL = CASE 
			WHEN @IndexKeySQL = ''
				THEN (
						@IndexKeySQL + QUOTENAME(ColumnName) + CASE 
							WHEN Is_Descending_Key = 0
								THEN ' ASC'
							ELSE ' DESC'
							END
						)
			ELSE (
					@IndexKeySQL + ',' + QUOTENAME(ColumnName) + CASE 
						WHEN Is_Descending_Key = 0
							THEN ' ASC'
						ELSE ' DESC'
						END
					)
			END
	FROM @WholeIndexData
	WHERE ObjectName = @ObjectName
		AND IndexName = @IndName
		AND Is_Included_Column = 0
	ORDER BY key_ordinal ASC

    
  --PRINT @IndexKeySQL   
    
  ------ Construct the Included Column string --------------------------------------  
  SELECT   
   @IncludeColSQL =   
   CASE  
   WHEN @IncludeColSQL = '' THEN (@IncludeColSQL + QUOTENAME(ColumnName))   
   ELSE (@IncludeColSQL + ',' + QUOTENAME(ColumnName))   
   END   
  FROM @WholeIndexData  
  WHERE ObjectName = @ObjectName   
  AND IndexName = @IndName   
  AND Is_Included_Column = 1
  ORDER BY key_ordinal ASC  
    
  --PRINT @IncludeColSQL  
  
  -------------Construct the final Create Index statement----------------------  
  SELECT 
  @FinalSQL = 'CREATE ' + @IsUnique + @Type + ' INDEX ' + QUOTENAME(@IndName) 
  + ' ON ' + QUOTENAME(@DBName) + '.' + QUOTENAME(@SchemaName) + '.' + QUOTENAME(@ObjectName)  
  + '(' + @IndexKeySQL + ') '   
  + CASE WHEN LEN(@IncludeColSQL) <> 0 THEN  'INCLUDE(' + @IncludeColSQL + ') ' ELSE '' END
  + @FilterDef  
  + ' WITH ('   
  + 'PAD_INDEX = ' + @IsPadded   
  + 'IGNORE_DUP_KEY = ' + @IgnoreDupKey  
  + 'ALLOW_ROW_LOCKS  = ' + @AllowRowLocks   
  + 'ALLOW_PAGE_LOCKS  = ' + @AllowPageLocks   
  + 'SORT_IN_TEMPDB = OFF,'   
  + 'DROP_EXISTING = ON,'   
  + 'ONLINE = OFF,'  
  + 'FILLFACTOR = ' + CAST(@FillFactor AS Varchar(3))  
  + ') ON ' + QUOTENAME(@FileGroupName)  
  
  --PRINT @FinalSQL  
  
  -------------Execute the Create Index statement to move to the specified filegroup----------------------  
  BEGIN TRY  
   EXEC sp_executesql @FinalSQL  
  END TRY  
  BEGIN CATCH  
   SELECT @ErrorMessage = ERROR_MESSAGE()   
   GOTO ABEND  
  END CATCH   
  PRINT 'Index ' +  @IndName + ' on Object ' + @ObjectName + ' is moved successfully.'   
    
  SET @CurLoopCount = @CurLoopCount + 1  
  
 END  
   
 SET @StartPos = @StartPos + 1  
END  
 SELECT 'The procedure completed successfully.'  
 RETURN  
  
ABEND:  
 RAISERROR (@ErrorMessage, 16, 1);
  
FINAL:  
 RETURN    
END 

GO

Per ojiNY's suggestion, I re-introduced index filters. Prior to that, it would have been possible to loose filters during the move to another file group.

I would love to see this also move the Unique Keys and Primary Keys, but I'm unsure how to check things in.
Here is the code I wrote to add Unique Keys and Primary Keys:

/* FIX all Unique constraints and Primary Key constraints */
DECLARE @SQL NVARCHAR(max)='', @Compression INT = 0, @Filegroup NVARCHAR(max) = '[PRIMARY]'
SELECT @Compression = SUBSTRING(@@Version, 22, 4) + CASE WHEN @@Version LIKE '%Enterprise Edition%' THEN 1000 ELSE 0 END
SET @Filegroup = '[Standard]'
--SELECT @Compression , @@Version
--how to enforce compression: SELECT @Compression=2016
SELECT @SQL += 'ALTER TABLE ' + a.SchemaName + '.' + a.TableName + ' DROP CONSTRAINT ' + QUOTENAME(a.KeyName) + '
ALTER TABLE ' + a.SchemaName + '.' + a.TableName + ' ADD CONSTRAINT ' + QUOTENAME(a.KeyName )

• CASE
  WHEN type = 'PK' THEN ' PRIMARY KEY ' + a.IsClustered + '(' + A.columns + ') '
  WHEN type = 'UQ' THEN ' UNIQUE ' + a.IsClustered + ' (' + A.columns + ') '
  ELSE ' unknown type '
  END
• CASE WHEN @Compression>=2016 THEN ' WITH (DATA_COMPRESSION = ROW)' ELSE '' END + ' ON ' + @Filegroup + '
  '
  FROM (
  SELECT kc.type, kc.name AS KeyName, s.name AS SchemaName, t.name AS TableName
  , (STUFF((SELECT ', ' + c.name
  FROM sys.index_columns ic
  INNER JOIN sys.columns c ON ic.object_id=c.object_id AND ic.column_id=c.column_id
  WHERE kc.parent_object_id=ic.object_id AND kc.unique_index_id=ic.index_id
  ORDER BY ic.key_ordinal FOR XML PATH('')),1,2, '')
  ) AS columns
  , i.type_desc AS IsClustered
  FROM sys.key_constraints kc
  INNER JOIN sys.tables t ON t.object_id = kc.parent_object_id
  INNER JOIN sys.schemas s ON s.schema_id = t.schema_id
  INNER JOIN sys.indexes i ON i.object_id = kc.parent_object_id AND i.Index_id = kc.unique_index_id
  ) AS a
  SELECT @SQL