In over 60 years of active deer hunting I have shot a few deer when they were running.
In fact we made a running deer shoot at our club and ran matches. The target 'deer' was on a pipe track and did not bound however it was good practice.
A favorite happened on the hill behind the camp in VT. I was hunting uphill in hardwoods and jumped a buck and doe which ran downhill off to my right. It was maybe 40 yards or so.
I remembered a trick to get deer to stop and I shouted "Maw" and they both stopped, turned and started back up hill. They don't go as fast uphill.
I got the buck with the first shot.
It was a happy but long downhill drag back to the camp.