For renters it is pushing out people with lower incomes. Not everyone (due to rent control in areas), but still quite a lot of people are getting pushed out I think.
For existing lower-income homeowners it creates an opportunity to get a really good price for their home and then move to cheaper environs. (aka Gentrification).
The remaining pre-existing homeowners are not necessarily going anywhere. Prop 13 means that their property taxes are not changing radically and living costs are otherwise on a less steep ramp.
Gentrification is a two-edged sword, for sure, but I'm not sure that anything can really be done about it. The people protesting the changing nature of their neighborhoods are in the same economic class as many of the people selling and moving away. A person from group A can't really force a person from group B to not sell their home.