The question of what happens if you move mid-year will be highly dependent upon your school system, and on whether you move within the same system or move to another one. On the summer move, I don't know of any schools that don't have strict policies on out-of-boundary attendance that would bar you from continuing, and many have policies that would allow them to assess you "tuition" in the thousands to tens of thousands per child depending on the length of time when your child was attending schools while living outside the jurisdiction.
Oh, and in some school systems, people watch like hawks for boundary violators so they can report them.