Given an Artin stack with a good moduli space, the morphism to the good moduli space behaves in many ways like a proper map despite rarely being separated. In this talk, I will explain two results in this direction. The first is the existence of generically finite proper coverings of the stack by a scheme, generalizing the fact that separated Deligne-Mumford stacks admit finite covers by schemes. The second is a strong version of the existence part of the valuative criterion of properness for good moduli space morphisms which generalizes a result of Bresciani-Vistoli for tame stacks. I will also discuss applications to the projectivity of good moduli spaces. This is based on joint work with Elmanto and Satriano and joint work in progress with Inchiostro and Satriano respectively.