Birds have nostrils and a trachea which connects to the lungs. Air interacts with blood in these parabronchi, bringing in oxygen and removing carbon dioxide and impurities. These merge together at the lung's outlet so as to allow air to eventually exit through the trachea and nostrils. Bird's lungs are firmly attached to body walls, and can't change size or shape during the breathing cycle. To force air through, there is a system of air-sacs, many of which have muscles arranged to contract and expand in a sequential manner, so that they keep bringing air in, forcing it through the lungs, then forcing it back out of the nostrils again.
Birds have a very high metabolic rate, and thus require a high breathing rate.
Some of the air sacs seem to accomplish this heat removal function, as well as being an inherent part of respiration. This contrasts with reptiles, whose body temperature is taken from their surroundings. If so, they haven't described it. We've discussed two of several features of birds that have, so far, been impossible to so describe. That looks like enough difficulty to meet Darwin's challenge.