We had the same problem for years when the neighbors had the barn across the road. Everyone in the county it would seem- thought it a great idea to drop off unwanted cats. I guess they thought they would become "barn" cats. Often times young litters were dropped off.
Every day I would scrape cats off the road that got hit by one of the many vehicles that travel to and from the farm down the road. The true/original barn cats that were around fed themselves, kept to themselves and went off to die by themselves. It was getting really bad for a while with different cats showing up on our door step and it wasn't fair to our indoor cats.
The old farm was purchased and the barns torn down. End of problem I thought. For a while at least.
A close neighbor had a female cat and couldn't be bothered to have it fixed. She would have a litter and most would die off because they didn't give mom enough food to support her needs for the kittens. When said neighbor moved away, they left the cat behind WITH a new litter of kittens! The were aware that she had the kittens before they left and didn't want to be bothered. We ended up having to round them up and have animal control come get them. All except for momma cat. She apparently found her way back into the trailer and got locked in. by the time we found her, she was wasted away and she was put down.
Sorry for the rant..... Animal control is more concerned with dogs, the shelter basically won't accept cats from individuals (so they don't become a dumping ground) they only take them from animal control....so up here, a vicious cycle exists that apparently no one cares to break. It is sad and unfortunate that cats are treated this way.
I stopped catching strays because of the risk to my own animals inside, cannot afford to feed them for days until someone can come pick them up (if they will). If the animal appears healthy, I discourage them as nicely as I can (spray water at them, throw snowball..)
If they appear ill (Neurological signs, obvious injury, malnourished)- I take care of the situation swiftly. I have domestic and farm animals to protect so I treat a sick animal as a sick animal- same as I would a racoon, skunk etc.
Not my favorite thing and still sucks to have to do it.