Breathing is important.
I know, I know: "Thank you, Captain Obvious!" But hear me out. Or...read me out...whatever.
In weightlifting, the Valsalva manuever is often spoken of. I'm probably oversimplifying, but essentially the Valsalva manuever is taking a big breath before you initiate a repetition, and holding that breath throughout the duration of the repetition. This causes an increase in intra abdominal pressure, and that in turn helps you keep your skeletal components in the correct biomechanical position to make the lift.
CrossFitters know of, use, and discuss the Valsalva manuever. But curiously, an equivalent technique for gymnastics is rarely, if ever discussed. However, it's just as important: try to do a max set of pushups, and note what happens to your breathing. As you fatigue, you probably end up doing something like this: 1) Pause at top of pushup 2) take a big breath and hold, 3) descend while still holding breath and aggressively push up. Something similar will most likely occur for pullups, handstand pushups, ring dips...pretty much any gymnastics movement in which you can reach failure.
Pay attention to your breathing, learn how to control it to optimize your work, and reap the rewards. Just make sure you let it out once in a while...otherwise you may pass out, and while that is amusing, it's ultimately counterproductive, since you can't finish the WOD.