Grat idea Dominatrix! The animal resources could be like small civilization only capable of building "invisible settlers". The "cities", could be an animal resource, and the so-called "invisible settlers" could cross the oceans onboard the ships of the human civs, just like horses came to america and rabbits came to Australia.
Yes, I think in a mod they had this, you could build a pig farmer in a city that had pigs and then take him somewhere else and he could propagate this resource.
However, if you think about it, Civ already has the mechanic for this in the game.
Let's say you have Potato resource out in Idaho. You don't need to "Seed" potatos through out your empire, you get the bonus of having this resource automatically in all the towns that are connected to it. So in a sense, horses were "Brought" to america because they were connected to ports that had access to horses.