Can you add in pics/vids of people running?
They just interest me.

Take these tips on balance.
First off, without balance in your replay, it's not going to turn out very well. You almost always need to keep your balance, or at least have some way to regain your balance quickly. You're not going to want to ko your opponent only to figure out after that have no way to get back up.
The easiest way to keep your balance is keeping your movements pretty fluid and staying on your feet. I have met many people who under-value keeping fluid movements because it's not their style. Almost all these people have no balance. Ensure that you don't do anything that will cause you to not be perpendicular to the ground. Doing a cork over your enemy is nice, but it's futile if you don't land on your feet, or you land on one foot. Not only will this destroy your balance, but it looks terrible, unless you do it properly.
Land on both your feet always. It doesn't have to be perfectly flat on the ground, but having two points where your weight is concentrated will help. Even just landing on the tips of both your feet will give you better balance then landing perfectly on one.
There is nothing wrong with pushing yourself of the ground with one of your hands if you do happen to lose you balance, whether by accident or on purpose. Lots of people say that you need to stay perfectly on your feet, however that depends on your style.
Finally, you're never going to want to make any sort of movement that throws you off balance, unless it's something good. Falling into a roll and rolling on the ground because you missed a high velocity punch would be realistic. Always gauge what you're doing and think about what will happen. If you realize that making your punch go faster will make you go off balance, don't do it. Moving your foot slightly forward to keep your legs spread out is always helpful. When you land a hard hit on your opponent, it's helpful to relax your chest and pecs a few frames before the hit. If possible change it to 5 or lower and time it so you relax your chest area about 4-6 frames before impact. This keeps your balance since your arm already has the momentum it needs to keep it moving, so when you relax, instead of plowing through and losing your balance, your arm will stop when it hits your opponent. It will still look like a solid hit, it will look more realistic, and keep your balance.

I've attached a short example of landing with both feet.
Also rather then quoting the entire post, which won't give the replays as well. Provide links to single posts.

Hello. There is a replay where Largekilla do a straight run. You can analyse his moves (by pausing the game and pressing E to see what exactly he is doing). I don't know if you could find a better run in Toribash.

The tips I could give are :

- More you put your weight in front of you (more your head is close from the ground) faster you will be. Thats mean if you wanna lose your speed (for exemple, transform a run into a walk), raise your body more vertically.

- Make little steps helps a lot than big steps. Avoir to contract/extend hips at maximum, do it only 20-30 frames max, then relax/hold.

- When you are running, its happens you fall on the side. Avoid to turn the chest. Don't forget to use your pecs to keep balancing (like in real life, invers pecs with hips - extend left hip = contract left pec) and lumbars help a lot too to keep balance.

- Use a lot fluidity (relaxing your moves after giving the starting impulsion) to avoid too fast moves that can easily ruin your run.

It's hard to understand what you mean in some parts of this tutorial, possibly give some replays for examples.