Like Riley said, you are best off trying to call animal control and let them handle it rather than try to move the animal yourself - if it's injured and in pain, you could end up getting bit. Whenever I've worked in either general practice or emergency, we've had people bring in injured animals all the time - they were designated "Good Samaritans" and not responsible financially for the animal's care if they were not the owners or did not want to assume ownership. If there were no ID tags, we would try to scan for a microchip. If an owner could be identified, we would try to contact them. If we couldn't get a hold of one, we still performed very basic emergency stabilization care - IV fluids, pain meds, wound care, etc. If the pet could be stabilized to the point where it no longer had to be in the hospital (and an owner still had not been found), then we would contact animal control and transfer it to them. If it was apparent that the animal had severe injuries that would require surgery and more expensive care to treat, and/or appeared that they were not going to make it, we would make the decision that the most humane thing would be to euthanize. We hated the idea that we could be euthanizing someone's pet without the owner knowing what was going on, but we also didn't want to see the pet suffer needlessly. We figure too that the owner would not want the pet to suffer either.
Once in a while, if an owner could not be found, but the injuries were minor or fixable, and the dog or cat had a particularly good temperment, one of the employees would end up adopting it and taking it home. One place I worked at had a really sweet young lab mix with no ID come in after it had been hit by a car - it had a fractured front leg that needed surgery to stabilize but otherwise was fine and healthy. One of the techs wanted to take her home if an owner was not found. We had a boarded surgeon on staff who agreed to repair the leg at a greatly reduced cost and then the employee would pay off the balance. The dog stayed with us 10 days total to give its owner a chance to come forward but nobody did (we placed ads too and notified the other clinics and shelters in the area). So doggy got the care she needed and found a new home.