During the first expiration, the air is moved from the posterior air sacs through the ventrobronchi and dorsobronchi into the lungs. The bronchi continue to divide into smaller diameter air capillaries. Two kinds of capillaries flow side by side, in opposite directions - the air flowing caudal to cranial and the blood flowing cranial to caudal (in general). This acts as a "countercurrent exchange" mechanism.
Blood capillaries flow through the air capillaries and this is where the oxygen and carbon dioxide are exchanged. When the bird inspires the second time, the air moves to the cranial air sacs. On the second expiration, the air moves out of the cranial air sacs, through the syrinx into the trachea, through the larynx, and finally through the nasal cavity and out of the nostrils.