Sometimes people just need to use some common sense.
If you want to get better at moving around quickly in your stance, actually practice moving around in your stance.
In the best book on boxing ever, 'How To Be An Asswhipping Boxer', one exercise Champ Thomas recommends is 100 meter shuttles forwards and backwards while in your stance.
Another exercise you can do is draw or tape a circle, with a circumference about midway inside a boxing ring or standard mma cage (depending on what you compete in), and practice circling while darting fully over one side and the other, facing inside and out.
You can measure it to push yourself as well, to get a certain number of line crosses with a certain distance circled under a certain length of time. You can also combine this with someone standing in the middle of the circle with a bag full of tennis balls, to throw at your head while you punch them out of the air (or dodge) to practice your hand eye coordination at the same time.
You can apply the same principle to heavybag work too. Make a circle around the bag (marking at the toe from the furthest away you can reach with your fingertips while balancing on one foot is a good 'dimensionless' quantity for all shapes), so you can practice circling while darting in to land one, two, or three punches, and darting out again (make it so you don't throw again until you touch both feet outside the line again to 'reset' it, so you actually get the workout and don't just crowd around the bag).
You can add a few wrinkles to spice it up as well; if doing low speed/low intensity/form work, you can mark off 5 or 7 equidistant sections, and make it so you never exit through the same section you entered to punch. When doing it at full speed, you can add a vertical line at one point in the circle, so that when you cross it with both feet, you reverse direction and start circling the other way (so you don't condition yourself into a rut). You can make a game of it even, to see either, how how much distance you can cover while hitting a certain minimum of punches/darts in a certain time frame, or, see how many punchdart sets you can complete while circling a certain minimum distance in a certain time frame, or track both at the same time to set PRs.
