Where I live, there is no structural vacancies at the low end of the market. In markets like that, unless you believe that UBI will cause a net reduction in the number of households, those landlords will experience an increase in demand, not a reduction.

In markets with structural vacancies (more units available for rent than households wanting to rent them), you might see of the migration effect you describe due to increased mobility.

(To be clear, I believe that UBI will cause an increase in household formation, not a decrease.)