Contracts work according to whatever they say and signed up for. Which could be anything. I totally get this notion that, "I would never sign up for anything that lets them adjust my pay or position", but there is some precedent here.
Off the top of my head, the Baylor President was removed from his role - not fired - after the Briles fiasco. He was "reassigned", before ultimately resigning himself.
An Oberlin woman's lacrosse coach said some things about transexuals competing in the sport that the university did not like. She was not fired. She claims she did not break any rules. Oberlin got a little wish-washy with their position, and ultimately She was reassigned due to "breach of trust."
Do I think these are exact parallels? No. But one possible takeaway is, if you really want someone moved, a way could be found.