Barefoot or not, this is important.
Run with your whole body.
Since you run with your whole body, strengthen your whole body.
I'll skip the legs, because it is obvious that running is a leg exercise. I'll work my way up from there.
You've got to have strong hip abductors and adductors. These are the muscles that move your legs away from the midline and towards the midline of the body. While running, as these muscles fatigue, they lose their ability to stabilize the legs. As a result, side to side movement can occur. When the goal is to propel oneself forward, side to side movement is undesirable (obviously).
Weak core muscles can be a real pain in the back. Strong core muscles hold the spine in a neutral position while running, if the spine comes out of position, the jarring motion of running can cause pain and injury to the back.
Strong upper back muscles help maintain good running form which is necessary for running economy, speed, and endurance.
Arms drive the running movement, if you doubt this, glue your hands to your side and run. Strengthen your arms so they don't wear out early.
A weak neck causes lots of head bobbing, pain and injury follow, as well as loss of running economy.
Running strengthens and builds endurance more in the prime movers of the running motion. As a result, many runners have imbalances between those muscles and all the other muscles that have supportive roles in running. It is important that we take the time to strengthen the rest of the body as well.