33 km is much longer than I was talking about (and lends evidence to the sibling thread's persistence hunting); I'm guessing most cavalry pursuits of routed infantry would've been between 1 and 10 km.

Taking world record 5k (5000m/12.58min) ~= 400 m/min

versus a top CCI run (6398m/9.97min) ~= 640 m/min

we see that at these shorter distances the horse has a distinct advantage.

https://www.youtube.com/watch?v=IEFeZuc3J3g