I would imagine that if a national UBI program were implemented, the amount would have to depend on the local cost of living, probably with caps. How they'd work that out, I don't know.

If that's not done, then UBI would encourage migration from higher cost-of-living areas to lower cost-of-living areas. That might not be a terrible outcome; it would probably lead to balancing the cost-of-living across regions as the populations change.