I was thinking a little while back about kobold thieves and goblin snatchers, and how susceptible my llamas would be to the local wildlife if I didn't build fences, and it suddenly hit me that there could easily be a class of thief that drags your livestock off the map, sort of like those stories of Sioux stealing horses as an act of bravery or somesuch.
While I could imagine goblins doing this and eating/riding the cows/horses/elephants/whatevers obtained thereby, or the kobolds being all coyote-like and making off with your stray cats and puppies, I think this would be a great thing to see if you had elvish neighbors:
Eilonwe Treefondler, Elvish Activist has liberated a Tame Rooster!
(Yeah yeah, they'd probably be called something else. Like 'Rustler'.)
Maybe you'd see this more if you antagonized the elves, or maybe they'd just do it all the time due to general elfbaggery. It'd certainly give elves an interesting boost from just being the easily-annoyed cloth merchants they are now. They'd be stealthed trapavoiders, much like kobold thieves. Guard animals would ignore the elves due to their 'friend to all animals' power, and then get 'liberated'. Maybe they'd open cages, too, just to be really sneaky, and get at your other animals. And next time you got elf-sieged, perhaps you'd have to deal with various 'Feral' animals brought back to their old stomping grounds.