The first one that leaps to my mind are 4e forced-movement (push-pull-slide) effects in combat; and trample/pushback rules in earlier editions. And traps, where an NPC actually sets them off just at the right moment. But those are physical effects, though still mechanical in nature.
A PC being mentally influenced without magic - Intimidate, Bluff, and Persuasion skills can try, if a DM has the stones to do it. But after that, there's not much.
1e has a bunch. Loyalty of henchman has a variety of mechanical triggers a PC can manipulate. Morale of opponents is often manipulated in games of 1e I run. Reactions of NPCs can occasionally be twiddled with through circumstance and previous knowledge of the character and motivations of the NPC.