The kiai represents karate's roar, the all-out strike. Try dragon breathing during your sanchin opening too. You should breathe during your kata between, not during movements. If sanchin is performed as a whole-body exercise, breathing after each movement adds a meditative feel that clears the mind. If the kata is performed for power or speed, then breathing less frequently makes sense.
Internal breathing is like a tea kettle that is building up steam.
As the pressure in the kettle increases, there will be a slow release of pressure though a small hole in the spout. With a rapid series of strikes, there will either be a release of several short bursts of air, or a release of a continuous stream of air out of the lungs. Exhale when delivering a strike or kick, and train yourself to exhale and tighten the muscles when receiving a blow. This prevents the air from being knocked out of you and prepares your body to absorb the impact. I find that when I focus I tighten all the muscles of my thorax and if I don't exhale, I can tell I'm straining somewhat, and that my breathing feels more natural doing it this way.