Ironically, a school-choice-ish program was tried in a metro area near me- the idea was to get some inner city kids out to the burbs, and some suburban kids into the city, to increase diversity.

The opposite happened- almost all of the kids from the city choosing the program were white, and all the kids from the burbs in the program to go to the city schools were black or Latino.

The end result (Which killed the program) was that kids and their families chose an increase in segregation, rather than diversity.

I expect it is almost certainly tied to socioeconomics here -- wealthier people can afford the hidden costs of the vouchers such as travel time, control of schedule, after-school fees and such, etc.