I imagine a numerous lifestock could also be a millstone around a players neck: if you want to move your tribe through unknown terrain, your lifestock forces you always to use a route that provides sufficient feed grains. Otherwise your animals unevitably starve. If that happens, or if you consume them or trade your herds away to be more flexible, you'd first need to make sure you find an appropriate substitute food source.
Also, when attacking a neighbor you have a dilemma: do you bring your herds with you, investing your livelihood in victory? If not, do you leave troops behind and attack with a smaller force, or risk another tribe making off with some of your herds while you're away?
-Like a farm, lifestock turns useless in wintertime unless moved to a warmer place (more grassland). During a winter turn when a lifestock unit is on "standby" (not grazing), it costs food instead of producing food. (ie. it costs 1 in winter, when it produces 4 in summer)
The way I envision it, herds would continue to provide food without pasture but their numbers would start to dwindle. I know nothing about the subject, but I imagine there is point in the lifecycle of cattle etc. beyond which you don't want to wait before slaughtering, so even during a month when your herd is starting to go hungry, you still have a "ripe" generation from earlier.
-Holding Lifestock could be a specialty of the horse-riding tribes like the huns.
Maybe herds could have a maintenance cost in horses? Except for horse herds, obviously…
Maybe a bit too fiddly, but something to consider.
I imagine the most difficult part of adding livestock would be the depletion mechanic. A simple way to do it would be to have each herd reduce the supply of its tile by one at the end of each turn. There would need to be some sort of buffer like units have, which goes up when the herd eats and down when it doesn't: the herd would die if it runs out, and could be split in two when it's full. Grazed tiles could have a chance to recover a supply point or two during wet seasons, in keeping with the theme of somewhat unpredictable map conditions.
However, since (as far as I can tell from the preview video) tiles only go up to three supply, there's not much room to tweak things. If all supply values end up being doubled and rebalanced from there, herding turns into a math problem that slows down the game, and could make things very difficult for the AI. At this point, I think the best solution would be to replace the minus-one-per-turn modifier with a single 'grazed: -50% supply' modifier that basically means "time to move your herd to a new tile". This way, the player could see which herds need to be moved each turn without having to look at any numbers.
Of course, the modifier would have to be governed by an invisible value on each tile measuring how much vegetation is left for grazing. I know invisible values aren't in the spirit of strategy games, but I think it would be enough for the UI to tell you how many turns your herd can stay in its current tile (or a prospective tile, when moving) before it becomes grazed or barren. As a bonus, implementing the veggie value would make it easier to replace the buffer with herd sizes, so that herds in fertile territory become larger and more productive, but graze tiles faster.