I think it gives realism to the game that the animals move. Animals usually flee from danger and migrate to other places. In all AOE the animals have a similar behavior (except in AOE-4, I think).
What I would do is add a new enhancement to the market, for example:
Salt the meat (For Europeans): Fishing (not whales and the like), cattle and hunting yield 20% more, hunted animals will not decompose when they die. (It would be unlocked with the advanced market card). I think it would also work for Asians and Africans.
Meat smoked (For pre-Columbian civilizations): Basically the same.
In the case of cattle, the limit of what can gain weight is increased, but the time it takes to gain weight is not extended.
In this way you can kill animals if you want them not to move far.