Rents are set by local wages (via mechanism of land rents)
> Rents might even decrease since any given UBI amount will go a lot further in a lower-COL area, which incents people to move out (reversing gentrification dynamics)
Can you identify any historical analogy to this claim?
History shows — across the board — people move toward higher COL areas as their ability to pay for COL grows.
Why would this be different?
It's difficult because the main driver is clearly towards increased urbanization, driven by the high productivity of urban jobs. But if you're planning to live mostly on your UBI and work less if at all, you won't care as much about that. There were several "back to the land" movements in the history of modern developed countries, and the early stages of succesful gentrification - often involving comparatively marginalized folks and highly mobile groups, such as artists and youths - demonstrate a similar dynamic that can ultimately lead to the flourishing of new urban areas as the stages of gentrification progress.
But this isn't all that related to the productivity dividend. That's more natural by setting banks to have a full reserve requirement. At that point new innovations would yield deflation, which would be harmful. The Fed would then have to step up and be the creator and distributor of new money. That money would then be the UBI, distributed equally to all.