Their morale is generally weak (-4 or similar) and gets weaker if they are

1) loosing
2) facing stronger enemy (I'm not sure but I think so)
3) being under fire, outflanked, attacked by more than 1 units, loosing with cav and many similar

To prevent it you should increase their morale
1) deploy them beetwen samurai units - don't let ashigaru be outflanked
2) support them with archer units and deploy your general close to them
3) as mentioned earlier build better dojo's, muster field, palace ans similar ( but its rather tip for later not for nearby battles)
4) use good generals to lead ashigagu - if I were you, I would use at least 4 stars