I've worked with it in practice a fair amount in the past. The ability to shoot on the move is definitely a valuable skill to have in your set, but like Rob said, you need to be able to do both. It may be that you are forced to stand and shoot through a port or other narrow gap, and there are classifiers or other "speed shoots" that keep you confined to a box.
When it is an option, it does depend greatly on the shooter's ability and the particular course design and there is unfortunately no "catch all" answer for one or the other, but generally shooting on the move is a good idea if you can. It may feel slower, and you may in fact be shooting a little slower, but if you are covering the ground and still putting the points on the targets, it usually works out to be as good or better than staying planted and sprinting.
Just with some basic drills you can determine if you shoot better on the move to the right, to the left, backing up, moving forward, etc. You can also work on whether it is better to shoot say a few targets on the move when all you can cover is 4 or 5 feet before you lose sight of them rather than a spot where you have twice or three times that distance.
Doing what you enjoy is always a good idea, but here is a simple drill that only requires two or three target stands and a couple of boxes. Set up 2 or 3 targets and use either two boxes, some sticks, etc, or just some other way to mark the ground clearly.
For the first test, put one box at about 7-8 yards and the other directly behind it at about 12-13 yards. Start in the back box, shoot each target twice, sprint forward to the other box and shoot each target twice. Run this a couple of times recording the time each run. Then do it again where you draw and shoot each target twice and repeat while moving between the boxes. Alternate targets, i.e. don't stack shots. Run this a few times so you can get the feel for the speed needed both in movement and shooting. You should be firing the last shot right as you get to the second box. After you get a few consistent runs in, compare times with the sprinting method. Note - the targets should be getting the same number of hits in each test. I'm not sure if I am explaining it clearly here. LOL.
After you have done this a bit, do it in reverse, shooting from the close box and backing up to the far box. This is good practice for backing up and keeping the muzzle within the 180, but of course please be careful.
Then move the boxes on a horizontal line approximately the same distance from the target, so you can repeat the above moving right to left and left to right in relation to the targets. There is so much you can do with this drill. Vary the distance between the targets. Vary it up by putting the box on the left slightly further back than the one on the right, and then vice versa, so the movement is diagonal if that makes sense.
We need to get together and shoot sometime outside of a match. Maybe when it cools off a bit we could just do a Saturday "practice session" and kick around some drills, ideas, and such and burn some powder. Just open it up to whoever wanted to come.