The resource system in Civilisation has always made me scratch my head a little when dealing with living resources. It's fine for, say, iron, which obviously has to be dug out of the ground in the correct location. Crops also make sense to be fixed if we take it as 'prime growing terrain' for those crops, thus allowing much higher harvests than elsewhere. However, I've always been confused why it is that I can't move animals, like, say, horses, to a different area of my empire but have to keep them pastured in the area my ancestors found them millennia ago!
Obviously, in the prehistoric era, it makes sense: you're just following the herd around, not attempting to breed it and move some of it to another city. However, by the time of the Classical Era people have surely figured out how to move a herd of horses from one city to the next!
I'd suggest that, for domestic animals, one should be able to build the (subdued) animal in any city which is able to gain the other benefits from that animal, unlocked when that animal is domesticated. For instance, after researching Equine Domestication, any city with a horse herd, or horses in the city vicinity, or a national horse breeder, would be able to make a horse unit, which could then make a horse herd in another city. Obviously, megafauna could be moved with Megafauna Domestication, etc. Obviously domesticated creatures without a specific tech would become buildable at livestock domestication.
Since the player still needs access to a creature to be able to make another herd, this wouldn't unlock constructions people couldn't make anyway. It also comes late enough that the production and food benefits from a herd are minimal. The only thing it does is allow a player to gain access to the relevant buildings in all their cities, and not find that any cities they settle after animals stop spawning are unable to get any herds. If you imagine what going out to create a new city may actually have been like, the settlers would quite likely would have taken large numbers of animals with them as they went, to make a living once they arrived. This would mimic this by allowing civilisations to move animals with, or after, the initial settlers.
As things stand, as I understand it, it's impossible for the Native American civilisations to gain proper access to horses even after they are discovered by the European civilisations, in a map like the GEM. It may be possible to get access to them indirectly, but there's no way to get a horse herd to anywhere in the new world.* This is obviously wrong, when you think about tribes like the Sioux.
*If there is, then please excuse my ignorance of the mod and ignore this point!
This would also deal with issues like this, and allow the Romans to build a Hippodrome in all their cities which are big enough. Given that according the Wikipedia they managed to make 61 circuses (which, it claims, are the Roman equivalent of the Greek hippodromes), it doesn't...really seem like something that only a tiny proportion of their big cities could make.