A Cooling Flush is effectively an active reload and is performed when you overheat your weapon and press
when the outer yellow marks are in the inner yellow gaps in the heat gauge on your reticule. The small outer ones move downwards so be ready.
You must do this 10 times total which will easily come through normal play through multiplayer.
Here's a video showing a couple Cooling Flushes, I did mess up the last one
Here is a screenshot during the moment also