What do migrating birds do during rainfall? Well, I guess that depends on the species of bird. Water-loving birds such as ducks can continue flying. Small songbirds can get weighted down and would need to consume more calories to compensate. Therefore, they are more likely to land and find shelter.
Birds can navigate around scattered showers. Widespread showers, however, are a different matter.
There is also a difference in behavior, depending on whether the rain is a heavy downpour versus a light shower. During the former, the birds need more protection. During the latter, you are more likely to see them performing normal activities such as feeding and attracting mates.
Spring showers create a good time to go birding, especially during migration. It just takes a little different gear such as a rainsuit, boots, and a cover for my camera and lens.
Great-crested Flycatchers are somewhat common in my neck of the woods. I would guess very few people ever see these birds but likely hear them.
It is easy for us to seek shelter during rainfall. Birds and mammals (non-human) must still go about their daily activities. Sure, the rainfall intensity will determine their activities. But they still must eat and survive. They don’t have the opportunity to stop by a local fast food place, eat, and take shelter from the storm.
During the next rainfall event, spend a little time outdoors watching wildlife as they go about their daily existence.