We have held on to a few dogs, but most don't let use near them and id rather not run them out into traffic trying to save them. Along stray dogs are pretty rare here. Cats, on the other hand, are common to see stray and we normally chase them off as both our dogs do not like cats and that will not end well.
One of my cats was a rescue from in front of our house. She was about the size of my hand and we have coyotes and fox that roam the neighborhoods at night sometimes. I’ve rescued a few dogs and helped them find their owners since i live in a massive metroplex and it’s likely that the animal is someone else’s. Cats are impossible to catch so I typically just leave them be though.