Movers are really pretty tough. It's easier to hit a moving target that is moving directly toward or directly away from you. A target moving laterally requires significant and steady movement on your part.
The difficulties of making such a shot depend on a number of issues, and these issues may behave in opposition to each other. At close range a moving target is going to require more swift and immediate movement on your part, over a greater arc. At long distances you'll be making less physical movement on your end, but you'll need to consider leading or "ambushing" the target by quite a bit (aiming for where the target is going to be when your bullet arrives at that distance).
At 70 yards you aren't going to need very much lead on a deer with a high velocity rifle round, at 400 yards you are. Frankly, the long distance lead is going to be VERY tough to pull off successfully in field conditions, especially when you consider that target speed is a guess. Quick math for my deer rifle gives a lead of about 2 feet at that 70 yard distance, and 18 feet at the 400 yard range. A bullet leaving the muzzle at 2710 feet per second will cover 70 yards in about 0.08 seconds, and 400 yards in about 0.49 seconds. Those lead figures are with an assumption that the deer is crossing perpendicularly in front of you at about 25 mph at a run. Clearly you won't be making calculations like this in field conditions, so you need to practice engaging moving targets are various speeds and distances (within the scope of what you're trying to accomplish).
Anyway, trying to track a moving target with a magnified optic can be very tough. Your best bet in this case is to keep your magnification dialed down to its lowest practical level. When I'm in the woods hunting I keep my rifle dialed to 3x. If a quick shot is needed I'll have a wider field of view, and an easier time acquiring the target. On the other hand, if the target is far enough away that I want 10x or higher magnification, I will have time to adjust up for those longer shots.
Anyway, that's my take on the issue.
Wow, so you come on here and admit that you are just as big as an unethical remedial as the first guy who shot at that deer. Interesting. That's WHY you don't shoot at running game - because YOU don't know how to make the hit. SMDH. Just un-REAL.
I don't see his question that way. He was asking about a specific instance in which he was trying to put down a wounded animal. That's a perfectly acceptable time to take such a risky shot.
Overall I agree with your idea that you should take shots at stationary big game animals under most circumstances. Moving targets are fun to practice at the range and in competition. And, in a combat situation it's sometimes worthwhile to take these shots, too. In the world of ethical hunting I'd venture a guess at saying 98% of hunters have no business shooting at a running deer.
Hunting is much like fishing: lots of grand stories are told around the campfire, and some of them are true. Most of the crazy ones were luck, and a select few people have the skills to reliably make the odd shots. Long range shooting is what I do for fun. It's how I spend 90% of my non-professional range time. As such, I've encountered quite a few stories over the years that simply don't make sense (ex: a coworker who claimed to hit a coyote at a dead run, off-hand, using his .30-06 and Kentucky windage, at 880 yards). I've also encountered quite a few hunters who are not skilled riflemen, and really only shoot 5-10 "practice" rounds per year. The "fish" stories create problems when they start to convince honest hunters that ridiculous shots are possible, if not normal.
Now, admittedly, there's a big difference between taking a shot at a deer you jump at close range in dense vegetation and taking a shot at a running deer that is 400 yards away across a bean field.
Regardless, I don't think the OP was asking this question with the intention of doing anything unethical.