UBI is slightly different because UBI would not be enough to live in an expensive city. So if you plan on living purely on UBI then you would need to move to the outer city or rural areas where COL is low.
I'd imagine that with UBI, the incentive to stay in a low CoL area increases and for a subset of the population, they will seek out a low CoL area. Then there will be the subset of the population that takes their UBI and goes to an expensive city for the "opportunities" and pay the higher rents for said opportunity. If it doesn't work out, they can take their UBI and go back to a low CoL area. Of course this mobility gets harder once one has a family.