Looks like your face map is set to PROJECTION MAPPING/FLAT MAPPING. It's projecting the face on to the head object much like a projector would project a movie onto a screen. If you moved the screen, the movie would not move with it. Instead, set it to PARAMETRIC MAPPING. This will solve the problem. You may need to use a RECT LAYER to get to get the face to the proper position but once you do it will move along with the head object.