Where do migratory birds go when it rains?
Obviously, as you point out, some birds, such as waterbirds (ducks, puffins, auks, sandpipers) are perfectly fine in the rain, as they spend much of their time in the water anyhow and have oil glands that waterproof their feathers. 

For other birds, rain can increase heat loss. For instance, rain increases metabolism by up to 22% in eagles (Stalmaster &amp; Gessaman, 1984), although the actual impact on average daily metabolism was 'negligible'. However, eagles were able to feed their offspring only half as much, and therefore had only half as many eaglets, when it rained (Elliott et al., 2005). (On the other hand, the authors point out that in the Tropics, eagles actually do better when it rains because there is more food available).

Migrating birds stop migrating, hide where they can, and wait until it dries up before they continue (Erdi et al, 2002); those that don't, often end up far away from where they should be and, eventually, dead. In short, some birds don't mind the rain (e.g. seabirds), others sit it out in misery (e.g. nesting eagles) and others hunker down wherever they can find shelter (e.g. migrating birds).