I'm vaguely aware that most infantry in the medieval period were peasant levies (if this isn't true, of course, let me know as well.) But how exactly would a medieval lord recruit peasants to fight in a war that might not affect them directly? Was recruitment for war ever a cause of unrest among the peasantry?
most infantry in the medieval period were peasant levies
This isn't true. /u/Rittermeister has an excellent run-down on the topic here.
Peasant levies during the period are really limited to a few specific contexts, such as the Early Medieval English fyrd or French levies during the Hundred Years War. In most of these contexts, the motivation for levies was mostly "the Danes/English are coming!"
The thing is that the person you're thinking about when you say 'peasant' may very possibly not at all look like what the typical non-noble foot-slogging combatant may have looked like in the Medieval Era. More can always be said on the matter, of course, so any further insight is welcome from anyone who'd like to post.
For the meantime, u/Rittermeister has three posts that examine participation in combat and why they'd fight. If you have further questions, we're always happy to answer follow-ups, and it's also likely that they may already have been answered, in which case I can dig up more posts for you.
Since u/Rittermeister makes mention of them in the first post, here's two relevant laws from the period: the Assize of Arms 1181 (the one that Henry II put out) and the Statute of Winchester - you'll want Item 6 for that. I'd have linked the Assize of Arms 1252 (which would be the one Henry III made), as it's my favourite of the three, but its text seems to have disappeared from easy internet view.