It is not always that easy:
This paper shows that racial composition shocks during the Great Migration (1940–1970) reduced the gains from growing up in the northern United States for Black families and can explain 27 percent of the region’s racial upward mobility gap today. I identify northern Black share increases by interacting pre-1940 Black migrants’ location choices with predicted southern county out-migration. Locational changes, not negative selection of families, explain lower upward mobility, with persistent segregation and increased crime and policing as plausible mechanisms. The case of the Great Migration provides a more nuanced view of moving to opportunity when destination reactions are taken into account.
That is by Ellora Derenoncourt, in the latest American Economic Review. Of course this also has implications for immigration policy, and it helps to show why even a very strong pro-immigration view ought to recognize some limits on the process.