Here's the classic Eadweard Muybridge run photos. From 1887!
It's quite a smooth motion with big arcs, because smooth motion and big arcs take less energy.
Click to view attachment
Notice that the left and right feet are almost always moving in opposite directions... when one goes up the other goes down, when one goes forward the other goes back. This is not nearly as true in a walk where one foot has to stay on the ground until the other one is planted. The faster motion of the run requires that the body masses balance out more.
The foot strikes the ground a bit in front of the body, but leaves the ground a greater distance behind. The foot reaches an extreme forward point and then starts to travel back before it hits the ground. The foot continues traveling back a bit after it leaves the ground.
Of course, this is only from our perspective of a runner on a treadmill. In real world space, the foot is getting yanked forward by the body as soon as it is done pushing off.
In real life there isn't much up and down of the body. In animation you might want to add a bit more to suggest weight.