As the title says, can I train a cat or a dog to do this?
Dogs are easier to train than cats, short answer is yes it is possible to train dogs to catch kittens. Many breeds of dogs can be "soft mouth" trained. In this scenario the animal they are retrieving is handled softly so they are not injured.
But the number of issues, make it too problematic to use as solution. For instance how do you identify the kitten is stray/feral? It would be nearly impossible to identify a kitten as stray as opposed to a cat belonging to a neighbor. How are you going to explain to the neighbor when your dog keeps bringing their cat to you? Also the risk of injury to the dog is substantial. Neither the kitten nor it's mother are going to appreciate the kitten being taken away by a dog.
You can train a cat or dog to catch others, but the cat can be scared of the other, or when it hunts it down it is not trained to bring it back to you. Cats have sharp senses and are more good at hunting than dogs, which over the years have lost most of their hunting abilities. My answer is that you can train a cat or dog, mostly cat, to hunt down another stray one, but only if it wants to , don't force it to, if it is scared. The thing is the feral cat can have diseases and illness so it's best not to touch it. Still you need to know if the kitten is really stray or feral, you wouldn't want to kill a pet kitten. You can tell if it usually is seen wandering streets, no collar, dirty, and check if it looks like it knows what it doing, if it is used to the wild.