Rents reset to market for new tenants. While it's true some well-off people hang onto apartments while living elsewhere, this is really rare. Turn over is usually quite high. Rent control mostly effects small multi-tenant buildings and especially in-laws, where 1 or 2 hangers-on can really mess with the economics.
My mother lives in a 12 unit building in a very nice SF neighborhood. Only 2 people have been there longer than 10 years, and only 1 other longer than 5.[1] The building was recently sold for only $2 million. The low price is less because of rent control, more because of all the other regulations that make it very costly to renovate or rebuild, lowering the value of the property. That building could/should be replaced with something having twice the number of units, but it never will be. In theory it should be trivial to have a system where existing tenants could be relocated while preserving their rent, but that would only work if there were much more supply coming online creating more structural vacancies so people could be shuffled around efficiently.
I think rent control is an acceptable policy in so far as it directly addresses people's need for a sense of security that simply can't be met by promises of low housing prices in the future. It does come with a cost, a kind of tax, but taxes can be justified. There are much more costly housing policies than rent control. However, those policies unfortunately tend to coexist with rent control (often preexist), so it's really just a mess.
[1] Years ago there used to be 2 tenants who had moved to and lived in Marin, but kept their units because they worked in the city and didn't want to commute every weekday. One was a dentist. But they moved out years ago. The building had no dedicated parking, so that may have factored into their decisions to finally leave. Dealing with that kind of hassle probably worked well (or at least sounded better) when they were younger, less so once they established a life in Marin, even if they continued to commute into the city. And it's not like housing is cheaper elsewhere in the Bay Area, so the extra expense is difficult to justify even for well-off professionals.