Class plays a big role in why white migration is not considered a net good.

White migration does happen, and we call it gentrification, because it usually means wealthy young (mostly white) folks moving into poor neighborhoods on the edges of "cool" cities, causing rents to rise dramatically over a short period of time, and pushing out longtime residents who are too poor to afford to continue to live in the hip new neighborhood with fancy bistros and cat cafes (yes, there's really a cat cafe...it's in East Austin on the site of a business once owned by a Hispanic family whose business was bulldozed).

Integration almost has to go the other way, or it is more of a disruptive force, and a force for further damaging communities of color. Race and class are so closely correlated in the US that you can't talk about one without the other on most issues. In this case, it means that white people moving into a black or brown neighborhood are often displacing black or brown people who can no longer afford to live there, because wealthier whites have driven up rents (and home ownership among black and brown people is much lower than ownership among white people in the US, so that doesn't provide a very good buffer to protect those black and brown neighborhoods from being "flipped" into white neighborhoods).

In short: Integration is a good goal. But, without some care, moving white folks into black and brown neighborhoods can be harmful.

Assuming a simplistic model where households swap locations (without handling population growth and new housing), a black household would have to move for every white household that moves.

Exactly - that's how I got the numbers above. 50% of the black population = 6% of the total population = 8.3% of the white population. My point is to assume self-interest on the part of the publication; it may or may not be representative of deeper unconscious bias in the population (there's no rule against both self-interest and unconscious bias being true), but a simpler explanation is that the Brookings Institute wanted to make their finding sound more dramatic.

Incidentally, phrasing it as "8% of whites need to consciously seek out diversity to end racial segregation" makes the problem seem eminently solvable.

It seems reasonable to give the 50% statistic, because it may show why blacks may be more reluctant or unable to participate in this mass movement. It's easy for a white person to say sure, let's move, given that there's only an 8% chance of any specific white person having to actually move.