Then it's even easier to move. If the polity doesn't want to zone for more housing, they've chosen to expel people who can't pay more than $X for rent. They'll feel the pain when there's no one around to make their coffee anymore.
There's no such thing as unlimited demand for luxury housing. Build enough and the rent will fall for the existing housing stock.
Moving requires finding a home somewhere that they can afford. The fact that they’ve been subjected to rising rents rather than a financial windfall like nearby homeowners… makes that far more difficult for them
They could have bought. Instead they took advantage of the optionality afforded to a renter that can up stakes and move at will, without being tied to a mortgage.
If they couldn’t afford to buy then they got to take advantage of a locale they desired at a lower price, another benefit of renting.
(While it would be nice if anyone could live — and own — anywhere they like without worrying about price or competing demand from the billions of other humans around, that’s so far away from reality it’s not worth discussing.)