I just used my fledgling Shimazu bow cav force effectively against some Chosokabe.

It was six units of Bow Archers, one general and one light cav versus 2x yari ashigaru, 2x katana samurai, 1x light cav and 4x chosokabe bow samurai

I thought this was a good test because the chosokabe bow samurai are among the best foot archers readily available in the game.

I used fire-at-will and swooping crane whenever I got in range of the bow samurai, using woods to disrupt their firepower and attempting at all times to bag them (ie. encircle them so I could benefit from enfilade fire and ideally at their unprotected rear).

I got pretty good results here. Each time they rushed a unit of yari or katana at my line I just fell back out of range of the bow samurai, turned off swooping crane and peppered the infantry unit with arrows until it was wavering, then I charged my general and light cav and shattered them.

So it seems bow cav are effective in a shoot-out against foot archers if used correctly. Having said that I was 6 vs 4 (although I used a lot of time and arrow on infantry units when I could have been encircling their foot archers, so it may have been balanced in that regard).

I have the replay if anyone wants to see it. It's not perfect (be careful of the dismount button, it's right next to swooping crane!) but not a bad example of HA tactics against a mixed force.

One other observation is that whatever penalty HA used to suffer in auto-resolve in previous games seems to have been reversed: my result for the fought battle was a close victory and I lost 300 men to their 650. In AR I lost only 39 and wiped them out!