Here’s a nice example for how run-ins work! This video is already edited down to the relevant parts, Ron Edwards
So, Adrian Neville (muscle-y guy) (babyface) and Tyler Breeze (carwash pants) (heel) are wrestling for the NXT title (which Neville currently holds). Tyson Kidd (another heel) shows up (in WWWRPG terms, his player just narrates that he comes out to ringside, I’d say) and Neville kicks him in the face – this is just narration, fine. Kidd and Neville have a good amount of Heat, you can hear the crowd pop when he gets kicked!
Kidd waits until Neville goes up for a big aerial move, then interferes! When he pushes Neville off, you hear the bell ringing – that’s the DQ, Breeze is disqualified, and thus loses his chance at the title, cuz Kidd interfered “on his behalf”. Kidd and Neville brawl.
Then the vid cuts to Kidd and Breeze yelling at each other, where you hear Breeze yelling “what are you doing, I had him!” It devolves into a general melee, until Kidd and Breeze team up to start putting the boots to Neville, which is when Sami Zane (babyface) runs in (at this point, not using the Move, I think), and they restore the face/heel balance.
This match led to various combos of these four guys in tag and singles matches, and then a final “Fatal Fourway” match where they all competed for the title.