For any shot, on still OR moving targets, if I am confident I can "make the shot", then I will usually take the shot. By "make the shot", I don't mean "hit the deer", I mean make a center shot on the vital area (i.e. hit a 4-6" target).

Every situation is different, so many factors must be considered. Speed of target, path stability (straight path vs random), range, direction of movement (straight away, towards, straight across, etc), number of game animals, stability of shooter, etc etc all come into play. Picking out a trophy buck out of a herd of deer bounding away over a wheat field is much more stable than making a shot on lone buck over rough terrain through woods. For stable-path targets, the "lead" is a quick calculation, but for a randomly moving target, the shooter should either wait for the target to reach a pre-determined position (i.e. a clearing, or gap between trees), or wait until the target has slowed down or the path has become stable.

Counter-intuitively, moving shots at 200yrds are often more simple than a moving shot at 20yrds. A deer running perpendicular (straight across) at 10mph at 20yrds is 880MOA per second. So basically, you "follow" a 10mph deer at 20yrds you would have to turn your body/rifle at a rate of about 2rpm. (10mph = 14 2/3fps = 176 Inches per second, @ 20yrds, 0.2" = 1MOA, so 176"/s = 880MOA/s = 25sec per revolution) At 200yrds, however, the turn is 1/4rpm.

I've taken shots on running deer, coyotes, and hogs. With practice it can be a high percentage shot, but an off handed shot on a randomly moving target is something I'm happy to pass on.

I practice on rolling targets. I cut 12 and 14" plywood disks then glued 2" rims for added momentum (14" solid plate with a 2" wide, 14" diameter "doughnut" glued to the back of the rim). I used an angle iron track to guide the rims down onto a carpeted track. A stop pin tied to a string holds the target in place until the shooter is ready. Bowling pins or plates swinging like a pendulum from a tree also work.