Moral is like a switch--men are either fighting, or their running. One thing I remember a seasoned vet pointing out in a post a long while ago: if a unit routs, all the men in that unit are instantly no longer fighting. Obvious, but significantly overlooked in many players' approaches to TW battles. This is better than slowly defeating a unit by killing its men. I've noticed that many of the really talented players seem to focus on causing units to rout en masse, as this will win a battle much faster than defeating your opponent by out-fighting them. Clan teammates are particularly good at this as they practice it.

Ashigaru routing through samurai won't affect the morale of the samurai soldiers very much, but samurai routing through samurai will affect the morale of the samurai who are not yet routing in a negative way. Samurai routing through ashigaru (both yari and teppo) will definitely affect the ashi morale in a negative way. Many players use low honor cav archers (h0-h1) very effectively (I love them) BUT, they can cause major morale problems when they route through your other troops, as they are samurai (and they do rout often if you don't manage them carefully ...).

There are a lot of things that affect morale, but the big ones are a unit getting flanked or attacked from the rear (particularly by a cavalry charge), or being fired on by guns (-6; this is a key reason why guns are so useful ...). Being outnumered in an area significantly lowers morale (this is why you should always bring 16 units in a MP battle). Your general getting killed will affect the morale of your entire army negatively (I seem to remember someone saying that in STW this is a permanent morale loss, whereas in MTW it is a temporary morale loss). If this happens at a critical point in the battle, it can spell your army's doom. Being fired on by archers or cav archers, or being charged by monks (if you're not Christian in a campaign) will lower morale of the targeted unit by 2. The rate of a unit's attrition obviously affects its morale.

As I said, many other things affect morale; these are just some of the ones most often exploited by players.