They do normally qualify for tenure and related benefits that regular teachers get. As for moving around, it can be quite stable (like teaching). However, one challenge is that it is quite common to be assigned to multiple schools. I have known SP folks who regularly are assigned to 3 schools per year, so they are working at 3 different sites per week. I even had one friend who was assigned 4 schools. While I think being assigned to two schools can be good in some ways (different task/duties/environment), 4 gets to be a bit much. With that said, I also have SP friends who are only assigned to one school.