I agree 100% with the transition training idea. The only part I would add is that you don't need to change stances... just walk forward and backward in a front stance... here you are transitioning from left front stance to right front stance...
If you watch people walk, many people walk by leaning their upper body forward, then catching themselves with their feet. You can see them tip forward slightly when they walk. Others will rock their upper body to shift the weight and lift their foot. If you train these people to fight, with high narrow stances, like you spar and fight in, it is very hard to get them to change to a more efficient way of moving. Get them into a low stance and it gets interesting. The lower the stance, the harder they rock their body and the quicker they have to move their feet to catch the fall. If you teach them proper low stance transitions, they learn how to pull with the forward foot, push with the back foot, how to move their center to stay on balance throughout the whole transition. They can eventually do it slowly and the rocking / tipping goes away. Now they move much better, stronger, quicker, faster and on balance... even when standing straight up.
Get a low enough stance, and its not possible to rock / tip your body forward enough to fall into that forward step. You have to take the step properly.
(and yes, this also develops muscle...)