Here's a post from u/Searocksandtrees that collects a lot of old links on this topic.

Something I didn't see discussed in those links (though I haven't read the last couple yet) is quite how much skill is involved to shoot a bow from a horse, particularly as stirrups and complex saddles hadn't been designed yet. The Assyrians' early cavalry worked in pairs like a chariot team without the chariot, where the 'charioteer' controlled both horses as the warrior shot his bow. Here are two pictures of a relief that shows this. This arrangement didn't last very long (as far as I can work out the relief is from the reign of Asurnarsipal II (668-627 b.c.)) but it's an interesting example of transition.