I've taken a shot at two running coyotes.
One was moving across a draw below me at about 80 or 90 yds. I shot, and missed. He kicked it into overdrive and started up the other side. I shot again and missed just low. As he went up the slope further, I had time for one more shot, and rolled him. I vaguely remember having the crosshair somewhere in the head area as he went up the slope. The bullet caught him through both front shoulders, dead before he stopped rolling. Since I was getting set up for deer hunting, I had my rangefinder with me. It lazed at 276 yds. Luck? Probably quite a bit, but I WAS trying to hit it, and did so, after two very close misses.
The second one was probably about 200yds out trotting across a bare field downslope from me, moving left to right. I miscalculated the drop on my 6.5 Swede and held just above his ears. The bullet kicked up dust just over his shoulders, and he hightailed it... didn't get another shot.
I came away with the knowledge that bullets get to the target a LOT quicker and flatter than I had believed. If you learn your load's trajectory, you can make a snap decision as to where to put the crosshair. Then it's simply a matter of swinging with the animal and squeezing the trigger.
My general rule is to NOT shoot at MOVING animals, if I can help it... much less RUNNING animals. I've been griped at by a friend when we got into a herd of hogs... he's yelling "shoot them", and I'm trying to pick a shot at a non-moving pig. (almost non-existent) Sometimes a running shot is all you will have, so practice helps.