I tend to find the training for walking and shooting best suited for those in an aggressor role such as military or Law Enforcement professionals who must take an aggressive approach to engage threats. The only situation I could see as being valid for those needing to walk and shoot is in an active shooter situation where your cover may be in front of you but you feel that you may have the time to take a shot with little risk of falling victim to returning fire. But i think that if you have this kind of time, i would just get into a low profile position and take a concentrated shot. If that does not give a satisfactory result, then rushing to cover would be the next move, not walking to it and putting a bunch of misses downrange.
Now the application of "moving" and shooting is a different animal. This means you are in motion but without a designated speed. This could be bear-crawling and shooting, or even shuffling in one direction and shooting. In most close quarters engagements, you aren't going to "walk" backwards while doing a tactical turtle isosceles stance and engaging a target touching the muzzle of your pistol. In close quarters, you want distance between you and your attacker, and this is best accomplished by shuffling or doing dashing movements, skipping laterally to move out of the line of attack. I recommend practicing in this manner instead of a "combat glide" that you see military personnel doing when advancing toward a target. If your going to move, be quick about it. If your going to shoot, be accurate. But don't pretend that you can just walk to get wherever your going. It isn't practical or realistic, and it is just another robotic lesson that you dont need in your toolbox.