Freedom of movement in the EU is conditional on you contributing to the economy; if you move just for the benefits, you will be kicked out (in theory)

What about asylum seekers and war refugees?

Not citizens, they don't have any freedom of movement to start with.

Most of them will not be able to work unless, or until, they're granted work visas.

The higher UBI areas would also be higher cost of living areas so in the right setup there would be little incentive to move just for UBI benefits.

But you would get to live in a higher cost area, at no cost to you.

What is the motive to live in a high cost of living area? Usually the two biggest reasons are economic and social opportunity. Economic opportunity is irrelevant in this instance if UBI cancels out in the two locations. Will there be enough social opportunities for people to leave their friends, family, and home country to move to a higher cost of living area?

The exact same is already true for minimum wage and salary in general, and while some people do migrate, there has arguably been no society-ending migration wave due to this reason. This is a common trope that has no basis whatsoever.

Although you can move freely to any country, to recieve benefits in a most countries you need to have had a job for 3 months before you can claim any benefits.