On the otherhand one could arguably provide a counter example in the form of white flight from the inner cities. An increasingly large and politically stronger black population tended to result in said aforementioned whites fleeing, mostly to the suburbs to escape "crime and lower housing values". Another thing to keep in mind is the fact that moving from one country to another is a whole lot different from moving from one state to another.
If it occurred on a municipal level why not on a statewide level?
It's not that it wouldn't happen at times, it's just that it wouldn't happen on a major level. The big ones would be bordering counties (usually rural, move to the next county over), especially those bordering major cities. The obvious ones would be Charlotte, NC and Memphis, TN. Maybe also Chattanooga, TN but I don't think Georgia would end majority black--since that's the case, add Savannah, GA, Columbus, GA, and Augusta, GA. Florida would be majority black before tourism/retirement takes off there, so add Pensacola, Tallahassee, and Jacksonville (not bordering but very close). It could go the other way too--Huntsville, Alabama could have development go north into Tennessee (perhaps about 40 mmiles northwest to Pulaski, Tennessee, birthplace of the Klan). Some of those cities (i.e. Memphis) might not be so good if you're a racist, since they'll be at least as black if not more, and earlier, than OTL. In any case, Southern cities would be much more black than OTL, since the "Great Migration" would be to Southern cities (we do have to ask ourselves why Southern cities are more attractive than Northern cities TTL when that wasn't quite the case OTL).
But on the other hand, if you live 100-200 miles to the nearest "white" state, would you really uproot your entire life just because you're that scared of the state government? I don't really see the a lot of the factors at play here other than the whole white people being scared of black people. Some people might. The majority of people wouldn't, leaving a phenomena which is only a fraction of what white flight did to cities and their surroundings. So in the end, perhaps mostly near bordering counties (limited effects), as well as provoking the development of exurbs in certain areas. Oh, and mostly halting the expansion of certain urban areas--for instance, Memphis would not expand south into De Soto County, MS, and if it did, the suburbs would be mainly black instead of white as in OTL.