When birds are migrating, they are working their bodies to the max. If you're lucky to see a large migrating raptor fly by at eye level, take a close look at the mouth area, and you'll see that they fly with their mouths open, to keep them cool as well as bring in enough oxygen for all of the hard work they put forth.
Even cooler (pun intended), birds do not sweat through their exterior like mammals do, but "sweat" ever so slightly through their throat. The process of sweating allows air to wick away moisture/heat from the body, allowing for a cooling effect to take place. When birds are overheated while at rest, or during their long trips within migration, they will gular flutter - panting to cool themselves via the moist muscles in their throat! Pretty cool, huh? :)