Sometimes, a Pet Gnome appears in your battle. It’s the one that looks like Burt Reynolds.
If you defeat the Pet Gnome and win the battle, a pet rescue is triggered for your guild. It lasts for an hour. BUT. If there is already an active pet rescue, you just get some pet food.
Protip: if there’s an active pet rescue, but it’s ending in 5 minutes, you can choose to sit in combat and wait to win until the pet rescue is over. The game decides if a new one starts when you finish the match, not when it starts, so this will let you start a new pet rescue.
So most guilds have a dedicated discord channel for pet rescue alerts, used both to help people know when a pet rescue is active and to judge how long one might have to wait with a gnome to start a new one.
In short: the more combat your guild does, the more pet gnomes you see, and the more pet rescues you get.