It should retain quality as long as you don't trample it, set it on fire, or run it into a rock
#got a question
tiny topaz
If you slowly drag it on a road, it should be fine (unless an npc steps on it)
Sometimes if you drag it at a full sprint and it hits a bump in the road and flies, that'll ruin quality too. Usually though I find as long as I'm not going full sprint, I can drag animals across most areas and be fine. But keep in mind this is a rockstar game. so sometimes the game just says "no" and ruins the animal.