Contributor: SWAG SUPPORT TEAM        

 I am interested in the source in Asm or TP to move a File from one
 directory to another by means of the FAT table.

All you have to do is use the Rename Procedure.  It isn't done via the
FAT table, but via Dos Function 56h.  The only restrictions are (1)
you must be running on Dos 2.0 or greater, and (2) the original and
target directories must be on the same drive.  The code might look
something like this:

Function MoveFile( FileName, NewDir: Dos.PathStr ): Boolean;
  f:      File;
  OldDir: Dos.DirStr;
  Nam:    Dos.NameStr;
  Ext:    Dos.ExtStr;
  Dos.FSplit( FileName, OldDir, Nam, Ext );
  if NewDir[ Length(NewDir) ] <> '\' then
    NewDir := NewDir + '\';
  Assign( f, FileName );
  FileName := NewDir + Nam + Ext;
  Rename( f, FileName );
  MoveFile := (Ioresult=0);
end; { MoveFile }