Hello,
My limited experience so far has taught me that if my troops are given "Charge" orders (and maybe "Advance"), they pursue the routed enemy when a battle is won. However, my infantry rarely catches up with routed men.
So, what I have been doing is to have some Cavalry or Horse Archers follow me from the beginning, otherwise if the battle is won, you can no longer issue orders. When the enemy is routed by infantry or other cavalry units, I pursue the enemy and run them down and when my units follow me, they also engage the routed enemy and kill them. This helped me level up my horse units fairly quickly.
I'm sure more veteran players also do that or have some other tricks up their sleeves. But I do agree with OP that being unable to pursue routed units is annoying. Maybe at the end of a battle, when it says the enemy is fleeing, we should have a button asking "Pursue enemy?" yes/no...or something like that.
Thank you,
APC