So here's the thought..
1 - Let us pick up baby animals like we can chickens et. al, but have them take up significantly more room (Say, a pumpkin for a baby pig, a 4x4 for a baby cow). So we can cart those little bastards around.
2 - Let us 'grease' them with Linseed Oil or some such. A greased baby animal will actively evade you, and will run just a little faster than 3 speed. Attempting to grab a greased animal will result in some form of check (RNG maybe? A brief combat exchange?) to see if you successfully catch it. Fail to catch it, it gets a quick burst of full run speed and the chase is on!
Yeah, it's just for fun, but village competitions for the win!