You rebuild the clustered indexes onto the new filegroup. You have 2 files in FG3. When you do a shrink with EmptyFile on one of them, the objects will be entirely in the other file on FG3. You then move them to FG4 as you would move any object to a new filegroup.

You cannot move a file between filegroups. You have to delete the file, then create a new one in the new filegroup and then move the tables over that you want on the new filegroup.