The D4 has the same AF system as the D800, I understand, and my D800 only lets you determine which af point to use throughout a shot sequence if you are shooting with Single Point. AFC S would appear in view finder. All of the other af point options 9, 21 what ever only permit you to select the INITIAL af point. After initial af is achieved the camera can change to any af point it thinks is best.
I would give the AFC S a try and see if that solves some of your problem.