Shooting at running game has fallen from grace with modern hunters and is often now criticized as being an example of poor hunting ethics. But for generations it was a well-accepted technique and was practiced by most hunters. Hunting writers used to publish articles on the subject.
Running-deer competition shoots were popular. For years Lyman had a logo of a running deer on its sights. Dizzy Dean said, "It ain't bragging if you can do it." I would paraphrase that it's not unethical if you can do it. All those old hunters knew they could hit a running deer, and any shooter willing to practice today will find he can do it, too.
It makes sense to practice, even if you utilize this skill just to finish wounded, running game. The most ethical thing you can do in such a situation is stop the animal from escaping. If you practice running shots you may be able to do that instead of standing there feeling, looking and acting useless.
Shooting a rifle at a stationary target is a fixed mechanical process that requires the shooter to execute a precise technique in a repeatable manner. If you try that on running game you'll be as frustrated and befuddled as Obama with a broken teleprompter.
Shooting at running game with a rifle requires a technique that's closer to shotgun shooting and it is more an art than a science. It takes practice. Rather than logically thinking about each step as you do with most rifle shooting, you must develop the ability to let your subconscious mind control the rifle. You still must be aware of the sight position, trigger control and all the thousand other things that make good shooting, but you must learn to let everything run on auto pilot.
This method works well when shooting at close targets, particularly in thick brush. It helps a great deal to have a rifle that fits you correctly and hits your shoulder with the sights lined up perfectly with your eye. Despite some conventional wisdom, low-power scopes are far superior to iron sights for this technique. Keeping both eyes open works best.
My grandfather taught me a variation of this when I started hunting deer 45 years ago. He said, "It's thick here in Vermont with lots of trees. If you try to swing your rifle with the deer, you will shoot a tree. To hit a running buck, pick an opening in the woods ahead of him and point your gun at it. Watch the sights, but keep looking for the buck with your peripheral vision. When the buck enters your vision, pull the trigger. If you miss, don't chase the deer with your rifle; get on the next opening, hold the rifle still and do it all again."
The rifle is kept moving the entire time; if you stop moving the rifle when the sight picture looks right, you will shoot behind the target. You must fire without hesitation. Hesitation will usually cause you to shoot in front of the target, as the gun will swing past the correct lead before firing.
As you learn these skills always remember there is an inherent danger in shooting at running game because the shooter tends to focus on the animal and block out the rest of his surroundings. It can be dangerous when the target is moving and the rifle is moving with it.
Imagine if the hunter is following a fleeing deer and just as he pulls the trigger his hunting partner's head appears in the scope. Both are focusing on the same deer and never even notice each other until it's too late. Never let your focus on the shot become so complete that you fail to realize where the muzzle of your rifle is pointing at all times. There is no animal worth the risk of shooting somebody.