Regarding shooting on the move, you have to have instructors who know what they're doing, and shooters who know what they're doing. I doubt it's possible to learn it properly on your own, without someone else's eyes who can accurately correct you.
If you do have those available, you can move fairly fast (think a fast-paced walk) and smoothly if you learn to do it right. It's more than just standing straight up and walking toward a target (whether human or paper). It's knowing when to squeeze the trigger and how to squeeze the trigger. It's also knowing how to move your feet and how to position your body. I'm not saying to you have to be in a full "combat glide", but you can make yourself more stable just by manipulating your center of gravity. I'm not saying that shooting on the move should be your default action, just that it is one of a number of options available in these scenarios.
Also, when referencing military IA drills, anything less than 50 yds is considered a "near ambush" and has an SOP for overcoming it. That, however, is based on having a squad (or team), with at least rifles, firing on an enemy element. Assaulting through is not necessarily the optimal option for an individual (IMHO, that is).
"All that is necessary for evil to triumph is for good men to do nothing." -Edmund Burke
"It is better to be thought a fool and to remain silent, than to speak and remove all doubt." -Abraham Lincoln
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