To INSTANTLY kill fleas, use baby oil. ANY cheap brand will work.
Use it on small puppies and kittens and apply using cotton balls or add it to a sink full of water and submerge them up to their necks. As you lift them out it will coat them.
Be CAREFUL not to get it in their ears or eyes!
We had better luck using cotton balls, plus, they seemed to like the “petting” and attention better than getting dunked.
Then simply pluck the little buggers off or use a fine tooth comb (or if outdoors, will fall off on their own). MUCH safer than chemicals. Then, if needed, simply bathe them with regular pet shampoo (or Dawn dish soap works GREAT), to remove the oil (may have to do it twice).