I'm assuming that you are not from the United States. First the federal government has no oversight into how the States organize their local entities. Second county organization has nothing to do with white flight as county organization occurred well before.

Each state is different in how it organizes villages, towns, cities and counties. This also reflects the state's unique history. For example we'll take my home state New York. New York has 57 counties excluding New York City, which just complicates matters. Each county and its boundaries are decided by the state legislature. Now cities, towns and villages separating from one county to join an adjoining one or form an entirely new one has happened. For example the towns cities and villages of the current Nassau County were apart of Queens County. Queens County became a part of New York City and those towns and villages who did not want to be a part of New York City were allowed by the state legislature to form Nassau. However, this had nothing to do with "white flight" at all since this occurred in the late 19th century. Indeed most county organization has occurred before white flight was a phenomena in the U.S. There are exceptions for example La Paz County in Arizona was formed out of the northern parts of Yuma County in 1983. However, it took time for this to happen mainly because there wasn't enough of a population in northern Yuma County to support a county government and afterwards Arizona made it much more difficult for counties to separate since then.

Anyway back to NY, each county can decide a method of governance whether it be a County Administrator, a County Legislature or a county charter. New York's counties were given delineated powers by the state legislature. Counties are municipal divisions that contain villages, towns and cities. Some county organization is historical, like in the lower Hudson Valley (their shapes, names etc). Counties cover large areas, Westchester County in lower NY cover 450 sq miles, Orange County cover 839 square miles, Essex County in upstate NY covers 1916 square miles. So as you see there is a big difference in county size. Mainly this has to do with population size but not always. Westchester is right above New York City with a population of just under a million while Essex has 39,000.

Towns cities villages can switch to another county. However, it is not easy to do. Since counties are merely administrative divisions it doesnt make much sense to switch. I'm not sure why there is some confusion as to why neighboring towns could be in two different counties things work differently in the U.S. and terminology is different. As I said county boundaries are historical, geographical, or are drawn to divide population administratively. There arent many instances where it matters if you are in one county or another. Like I said taxes, funding or maybe a certain law but otherwise there isn't much difference. If your town borders on Westchester County and you want to join Putnam County (which is right above) you would actually lose more than you gain. Westchester receives different funding from the state due to its high population and needs but it also doesn't mean a town can just decide to leave Putnam County and join Westchester. If a town or a group of people want to change something in their county they do it via the county government or state legislature.

Now New York City is a whole another monster. Due to its immense economic power and high population different rules apply to it. For example New York City contains 5 counties that we call also boroughs. New York County (Manhattan), Queens County, Kings County (Brooklyn) Richmond County (Staten Island), Bronx County. New York had for a long time been trying to combine the outlying areas which is another history lesson. New York is unique in that it incorporated Brooklyn (Brooklyn at the time was the third largest city in America), the towns and villages of Queens County, the towns and villages of the Bronx (which was a part of Westchester County, the Bronx was not a separate city when it combined with NYC but a collection of towns and villages), and the towns and villages of Staten Island in the late 1890's. However, New York City and its boroughs are not governed by a county government there are borough offices which influence policy, administer city services etc, but they do not govern.

I hope this helps and let me know if you need any further clarification!

So couple things. States define counties not the feds. So most of this will differ in a state by state basis. Now in states that started out as territories generally used the public land survey system. Which was much more uniform than the original colonies. Some states in the West like Texas and Louisiana are more based on other systems. But in many cases the counties are older than the cities. Now something else to consider is the structure of state governments can differ widely, the civil war and reconstruction influanced changes in many southern states constitutions.

So you end up with counties being often defined in the constitution of the state. So your question could be seen as asking why the Kansas/Missouri border isn't shifted over so all of Kansas city is in Kansas.

Now in some states the city is actually cut out of the county, this is more common in New England. It's more common for cities and counties to share jurisdiction.

So what does a city do, in my stae we have weak cities, essentially its a tax district. It provides police, fire, water, drainage, contributes to roads. The county provides police outside the city, and most importantly the state constitution requires it to handle the courts. This is why the city doesn't tend expand outside the county. So they only have to deal with one counties court system.

Outside of this we have other tax districts like MUDs that provide services like water/fire. In the 90s the city almost annexed a part of the teritory but they would have to of taken over the mud that carried that areas debt. These special tax districts aren't limited to one county, school districts famously cover many in sparse areas.

The cities can have their own laws(called ordinances here) so suburbs in neighboring counties often set them up, but often times these satellite cities doen't cover the same services and leave it to MUDs. But sometimes another small city will grow in size to fill the role.

City politics is complicated, we have enclaves and areas we've annexed than de-annax later. Counties are meant not to change. Part of this is we have vincinage requirements. When you break a law you have to be tried in the county the crime occured. This would be complicated if the county changed.