What happens to a person's facial implants over time? Do they move around?

Not usually. There are different types of facial implants and techniques. Most implants are designed to fix in place as they heal and sometimes there fixed with screws or stitches. Rarely an implant may shift or become infected.
No they should not. I assume you mean either chin, cheek or mandibular implants. All of them, if placed correctly should not move once everything is healed. Of course, in cases of trauma, something may occur to cause them to shift, but this is very rare. The majority of patients are extremely happy with the results from this procedure provided they have realistic expectations before deciding to undergo the procedure.
Possibly. Porous polyethylene implants are screwed into the underlying bone and are ultimately secured by tissue ingrowth. Because of this they are preferred over silicone implants which can move and can also cause bony resorption.