The point of the OP isn't that people are "moving," it's that they're figuring out ways to basically "segregate-in-place," the most obvious method for doing so being the creation of exclusive public and charter school catchments that allow people in diverse neighborhoods and suburbs to nevertheless ensure their kids don't have to go to school with the poor minority children.
Often times the worst offenders are otherwise "liberal" white elites in urban areas. Case in point is West Philadelphia - about 10 years ago or so the University of Pennsylvania partnered with the Philly school district to commit additional funding and programs to a public K-8 school near the college campus. Naturally, home values and rents in the school's catchment area skyrocketed, driving out the previous, poorer tenants and homeowners, making room for yuppies and Penn professors to live near campus and send their kids to a public school that de facto excluded almost all the children of working class and poor people of color that live nearby.
They in essence created a new opportunity for neighborhood kids that systematically excluded kids of low socio-economic and racial status, and displaced people in the process. How messed up is that? The kids of Penn professors are going to be fine regardless of where they go to school, but they get opportunities created and handed to them by public institutions.