My two cents worth: My dad started me hunting deer back in the 1950's. We were poor as church mice and needed the meat what with a family of 6 to feed. One day he let me go hunting by myself. Wow. That was big stuff. I was wandering up a hillside when a nice 3 point whitetail jumped up and set sail going flat out across the hillside. I saw when it jumped up that it was a nice deer but it was mostly out of sight in the brush until it got to a opening about 80 yards off to my right. I was ready with my peep sighted 6.5mm when he got there and I squeezed off a shot.
I missed of course and was surprised to say the least when I got over there and found a dead 3-point piled up in the bushes. (Trick question, should I have taken that shot?)
Over the years I shot a few more running deer though none were moving as fast as the first. Eventually, I made a bad shot or two and hit animals I tracked but didn't get. I felt pretty tore up when I wounded an animal and finally the light bulb came on and I realized you must be SURE WHERE YOU ARE GOING TO HIT THEM OR YOU DON'T TAKE THE SHOT. It would have been a lot easier if my dad had just told me this but that wasn't his style.
So... I'll tell you, bottom line, it doesn't matter what the game is doing or how far away it is, if you are sure, take the shot. For most people, including me, that means that running game is a no go... or game at long range in a big wind, and so on.
By "sure" I mean you gotta be SHOCKED if they don't fall down when you squeeze the trigger. If that is your reaction to a miss you are doing it right. If you kind of suspected that you could miss and you pulled the trigger anyway, I suggest you reconsider your whole approach to this hunting thing.
One more thing - back to the trick question. I was surprised I hit that deer. I had NO BUSINESS taking that shot. If you've got kids you are training up to hunt, do them a favor and don't let them find this out the hard way.