They save what can be reused, dismantle or alter what can be reused in a different form, etc. The Ghostbusters PKE meter shows up in They Live, Star Trek altered costumes and models as needed.
Some things are kept for historical value but these props were never designed for longevity. The original Alien suit is falling apart, for example.
Even if it is saved for historical purposes it may get thrown out or destroyed anyway if the new guy changes his mind- see the Phantom of the Opera original set being torn down to make space for a fucking roller coaster no one needed. Yeah I know a movie set isn't a prop but come on, why destroy history for needless shit? Props are always tossed to make warehouse space and no one hears about it. There was a time when the BB threw out or taped over actual film reels of episodes or movies to make space, which is why so many old Dr Who episodes were lost, and the entirety of Monty Python was nearly trashed by this policy.
Many end up in the wild, but without some sort of paperwork or other material to authenticate it, it's basically worthless. See half the shit that ends up on Pawn Stars that someone insists is genuine but can't prove.