Running is a sagittal plane activity, meaning you move forward and backward through space. Flexors and extensors such as your hamstrings and quadriceps act as primary movers to propel you forward toward the finish line.
Running also uses muscles in your frontal plane, namely the hip abductors (gluteus medius, gluteus minimus) and the hip external rotators, which act to maintain core postural control so you can move forward effectively and efficiently.
Endurance athletes with weak hips are easy to spot from behind because their not able to keep their hips level as they run. When they plant their foot, one of the hips drop instead of the right and left hips remaining level. The dropped hip will be on the opposite side of the body of the planted foot. So during a right foot plant, the left hip drops—and vice versa.
This happens because when the right foot is planted on the ground with the left foot raised off the ground, the left side of the body is cantilevered over to that side. Like a deck of a house cantilevered out over a river without any structural support directly beneath it (think Frank Lloyd Wright's "Fallingwater"), support to hold up that deck must come from the non-cantilevered side.
In the case of your body in motion, when your right foot is planted, the stabilizing muscles in your right hip (gluteus medius and gluteus minimus) do the work of holding up the left side of your body.