I don't really do any "training" for self-defense, per se, but I shoot a fair amount of USPSA. I shoot close targets while on the move with some frequency. I have learned that shooting at medium-range targets while moving is mostly a low-odds proposition, and rarely worth it. Every once in a while I'll sling a round at a full-sized popper at <50' while running if I have a chance to shoot it later or while running into the position where I'll engage it; sometimes I get a hit, sometimes I don't... but with steel, at least I know. I wouldn't waste the ammo trying to do it with paper, because I'd re-shoot it later anyway.
I don't even think about taking long shots while running. Nor shots at swingers or texas stars or other targets with lateral movement. I might shoot a close drop-turner while moving, but that's a different proposition... the aim point doesn't change, just the availability of the target. No tracking (much less leading) is required.
For the most part, though, unless a target is close enough to point shoot, you're better off moving hard to a spot from which you want to shoot, then shooting quickly and accurately, then moving again.
How does that apply to self-defense? Probably pretty closely, I'd guess. The difference would be in the value of misses. On the one hand, "suppressive fire" might keep someone from shooting at you. On the other hand, there's usually not a berm in the real world, and you're responsible for all those misses. It would be case-specific, but I'd generally say that you should only take self-defense shots you think you can hit reliably. And that's going to generally mean not shooting on the move at other things on the move.
In a combat situation, the calculus is probably significantly different.