This has become a fascinating topic for me.
I discovered that my breathing was best, during running, if I spent a few minutes before my run focusing on EXHALING as long and as hard as I could - apparently blowing out whatever unnecessary stuff was in my lungs. If I felt tired, or fatigued during my runs, I would take 10 long deep breaths, inhaling through my nose and out through my mouth.
Just in the last year, a new problem arose for me, regarding running and breathing - the source of which I only discovered in the last week - and though it is mostly specific to women, it shows the importance of proper breathing while exercising (and always).
For the last 6 months, I suffered from pelvic floor dysfunction (which is a long fancy word for spasms in the floor of your pelvis). I did some reading and it turns out that pelvic floor dysfunction is a more prevalent problem for female runners. It kept stating that "weak pelvic muscles" and the "jostling of the pelvic region while running" were he culprits. This sounded dumb to me - didn't make sense.
I just barely returned to running about 3 weeks ago, and after the first couple times, my diaphragm muscle seized up and to some extent triggered the pelvic floor symptoms again, and I was like WTH?
But then I ran across this intriguing article that actually explained how the DIAPHRAGM muscle is not only attached to the pelvic floor muscles, but it CONTROLS the functioning of those muscles. That means your breathing indirectly affects those muscles, as well.
So in the last 3 days I have started focusing on correct breathing (long "belly" breaths in through the nose out through the mouth) while doing ALL exercises (weights, swimming, running), and several times throughout my day, and my muscles have hardly bothered me.