I dragged home tons of animals...cats, dogs, horses...I even rescued a many wild bird, mice...you name it I "saved" it and then hopefully sent it on its way. But my parents were always strict about it being "my responibility." Meaning I had to pay for all veterinary costs, food, housing, anything...and if it was a puppy my mom made sure that I got up every 2 hours to potty train that sucker and "do it right...if you're going to do it."
Give your kid that much responsibility and see if she still brings the critters home
If she does everything perfectly...then I would just let it be.