I can instruct you how to run to make it possible to reload a MLer on the run, but I am not going to suggest everyone can do this, or anyone can do it over unknown, rough ground, as shown in the movie.
You use a "fax Walk", but run. It differs from how we mostly walk, which is upright, with our knees locking on the leg we thrust forward, so that we lean and then fall forward onto the heel of that shoe, and then shift our weight from the back foot to the front foot, as the front foot slowly lowers to lay flat on the surface.
With the fox walk, and "Fax Run", you unlock your knees, stoop over a bit, and you leave your weight on the rear foot while your forward foot is placed flat on the surface. Only when its fully down flat on the surface is the weight shifted from the back foot to the forward foot. The appearance to onlookers is that the person is gliding along, as there is NO up and down motion that is easily seen when we walk with locked knees. Indians used this Fox Walk, and Fox Run to cover miles of ground quickly, without the stress and pounding on the spine, and chest wall muscles that occur the way we westerner walk and run. W/o all the internal organs bouncing up and down, your body is able to be more efficient at breathing, and cycling blood and oxygen to the muscles in use.
The first time you try it, you will feel strain in your hips, knees, and ankles, simply because the muscles and tendons are not used to working that way. However, it is a much better way to walk, anytime you are walking over rough ground. By leaving your body weight on your rear foot, you can feel the ground ahead for sticks, stones, and debris that might otherwise trip you, twist and ankle, or do other harm, that can be easily avoided by moving that foot back, left or right, or sometimes further forward. It is the method of walking to use when going into the woods to hunt, and while stalking. It is amazingly quiet, compared to the noise you make when walking " normally!"
There are exercises you can practice to get better at using this walking/running technique, and get much more adept at covering rough trails and ground on the run, while still being able to use your upper body for separate movements. I have students put on blindfolds, and remove their shoes and socks. Then they follow a rope around a trailed course, feeling their way along with their feet, and hands, instead of Looking their way over the ground. They all report being much better in tune with their feet, and find they are less likely to step on stones, or sticks, or trip over things after doing the exercise daily over several weeks. Most adults can't remember the last time they walked down a trail in the woods, or even on the grass in their own back yards, barefoot.
It would be silly to think that you could just put on a loin cloth, or leather pants, and leather moccasins, grab a MLer rifle, and take off up a steep mountain trail reloading and shooting your gun on the run, without training yourself in the skills needed to do such a thing.