Sounds like yet another argument trying to deny the law of supply-and-demand, like those who quote the laws of thermodynamics and then claim they have a perpetual motion machine.
Instead of having to work for that first $1000/mo, it just...appears. Yes it was earned elsewhere, and whoever earned it won't miss it as much as the recipient will appreciate it, but since the recipient has done nothing for it, its local value plummets. Instead of some low-income people having $1000/mo for rent, now everyone has $1000/mo for rent - and basic prices will rise accordingly.
I won't notice much change for losing a sub-1% of my income, but those for whom UBI is really intended can go from really struggling to renting (even buying) housing. With limited housing supply, suddenly prices will rise.
While I'm a staunch advocate of "move already!" for solving economic problems, I don't see enough people moving - nor enough housing for them to move to - to cause "the price of housing on avg will plummet for UBIers".
Imagine the poor section of town. Many/most rent. Suddenly, with no additional productivity, everyone is getting checks for $1000/mo. You think rent etc prices aren't going to suddenly skyrocket? I grasp your theory that people can now move to more affordable locations, not needing to be near jobs - but will they really? before they can, rent prices just shot up, consuming their newfound income. Destination rental prices similarly increase - still relatively attractive, but higher than before since there is more demand and more money to pay for it.
Yes the supply of acceptable housing for UBIers increases, but those markets already existed and were already priced for a low-demand market; now you've "exploded" the demand - prices will rise, matching people who now have $1000/mo and few expenses.