No one might be entitled to that, but it's definitely helpful to the social health of an area if there is policy to ensure that the community stays somewhat stable.
I'm not sure that I agree unless we come up with some definition of "stable". Usually a neighborhood is healthy when people want to move into it, meaning it experiences growth, and it can lead to problems depending on how the growth is managed. But when a neighborhood stagnates and becomes known as a place that excludes certain classes of other people or makes them "unwelcome" then the neighborhood's only hope of survival is to contain a bunch of already wealthy people who entrench the existing classism or ageism or racism, etc.
When the neighborhood is working class and props up "stability" (in the form of not letting new people in unless they are 'like us'), it's a recipe for disaster.
This is an extremely hard problem to solve. I had a good family friend who lost her job as an elementary school teacher because a neighborhood rapidly gentrified and people moving in did not have children. Over about a 2 year period, rents rose and school enrollment dropped severely and the school had to cut staff.
But I also have a friend whose car was vandalized repeatedly in a neighborhood of Pittsburgh because he was associated with a tech company there that was popularly blamed for gentrification. He felt scared living there, which he should not have to feel no matter what the reason (i.e. he does not belong to any minority groups in terms of race, etc., but deserves to feel safe at his own home like anyone else does).
The idea of "stability in a neighborhood" is hard to pin down, because things can range from outright xenophobia to thoughtless gentrification, and market economics usually just acts like throwing gas on the fire no matter which end of the spectrum you're in.
