I've always thought it was funny how whites moving from the city to the suburbs is bad (white flight) and whites moving from the suburbs to the city is also bad (gentrification).
It isn't change per-se that is usually the problem these days, but the rate of change, and sometimes the rate of change of the rate of change.
Legitimate complaints (e.g. this will mess up traffic) are fine. "Neighborhood feel" complaints (e.g. this building is too tall, it doesn't fit the neighborhood character) should be met with "thanks for your feedback, but that's not a good enough reason".
Of course if there is widespread consensus among voters on how they want their city to be developed, fine. But right now, the current system allows a handful of very vocal opponents to development to derail new housing.
That's why doing housing/zoning policy at a much higher level, ideally statewide, would be much better. Japan has that (for their equivalent of states) and you don't see these distortionary effects. Then you'd see the development happening where it really truly does make the most sense, and as a side effect gentrification would be lessened because not as much new construction would be happening in farther out marginal neighborhoods because the demand would be satisfied closer in. Incidentally this also has positive benefits in lower commute times and less environmental impact.
In that case, vulnerable populations in gentrifying neighborhoods were no more likely to leave than equivalent populations in non-gentrifying neighborhoods.
For me, the obvious mechanism is that, well, it's true that people who leave may often blame gentrification / increased costs for their departure. But on the other side of it, without gentrification people may still have left, but for different reasons: to get away from crime, to go somewhere with more amenities. It's possible that these two tendencies somewhat balance each other out.
The question is which of these is worse for the people living in the area.