Maybe they can bring their resources with them too, and the poor schools can have things like lead paint remediation, honors classes, and extra curricular activities.
I like this idea!
Resources are mainly property taxes. Unless they're building a new house whose value is significantly greater than the median for the area, families gaming the system aren't likely to have much of a lasting impact on the district.
Or it could end up like some Asian countries with a large afterschool tuition industry. I guess at a minimum you don't want the kid to get shot up at school though.