Birds fly south
Only the ones that have access to the current global average surface temperature figures.
The rest of them will just respond to local weather conditions.
What makes you think that they haven't done so already. When I was young, mockingbirds were strictly southern. Today, they are common north of the Mason-Dixon line.
Yes I’ve heard they are moving because the rent is cheaper.
It's already happening.
they will move closer to the equator
If they are adaptable, maybe.