Does the aerodynamic center move when flaps and slats are deployed?
The aerodynamic center does not move when plain or camber flaps are used. A plain flap will only change the camber of the wing and, therefore, will only affect its zero-lift pitching moment. Flaps have their own aerodynamic center, and their lift change should be summed up in the flap's aerodynamic center.

Note that the aerodynamic center of the aircraft is used for the lift force change resulting from a change in the angle of attack while the flap aerodynamic center is used for the lift force change from a change in the flap's deflection angle. With flaps deployed, a change in the angle of attack will still cause a lift change which acts on the aircraft's aerodynamic center! Only the constant pitching moment around the aerodynamic center will change with flap deflection, and in order to calculate this moment, assume the lift change from flap deflection to act in the flap aerodynamic center.

The location of the flap aerodynamic center changes with the relative chord of the flap. For a flap approaching a relative chord of 0, the theoretical flap aerodynamic center is at 50% of the wing's chord. With increasing flap chord the flap aerodynamic center moves forward proportionally with the flap's chord increase until it reaches the 25% chord point for a flap of a relative chord of 1 (a full-flying control surface, in other words). See Figure 46a from chapter 5 of Sighard Hörner's Fluid Dynamic Lift for proof.



Flaps which increase the wing's area will of course shift the neutral point in proportion to the change in the wing's chord effected by the flap.

Slats may be viewed like a slat-sized wing followed by a wing-sized flap and an incidence change equal to the slat's deflection angle.