In Pacific Rim, the Kaiju emerge from The Breach and show up on the radar. They are identified and the Jaeger(s) deployed.
It was my understanding from the movie that
each of the Kaiju that comes through is eventually killed; they are not escaping back into The Breach and then returning again. It is mentioned that the Kaiju seem to be evolving, each one different from the last, furthering the argument that they are not the same ones returning.
When Choi identifies the Kaiju, he refers to them by their "codename" (Leatherback, Raiju, Knifehead, etc) - but who came up with these, and when?