At deer and elk; very very rarely do I shoot at them as they are moving. This is mostly due to the majority of my shots being in the 4-5 hundred yard range, on average. If I happen to be under a couple hundred yards, I will reassess. It's all about low percentage shots vs high percentage shots for me.
The flip side is that I've shot (and shot at) a bunch of coyotes who were in various stages of unassing the area...