Except everywhere is a climate danger zone.
Some places are being hit harder than others. All else being equal, people should move to the places being hit the least.
Wouldn’t a side effect of that, if done en masse, be property values and rent ballooning out of control in the safer areas?
A side-effect? No. You’re just describing climate-safe housing being more valuable. It’s always been more valuable.
In a functional market system, higher rents will result in more housing construction in those areas. I’m not delusional enough to think that the housing market is functional, but that’s a can of worms that will exist regardless of the climate problems or not.
Or to rephrase a bit: yes, if people all try to move to more climate-safe areas, then we’ll need to build more housing in the climate-safe areas for them to move into. Obviously.