Not all products that are 'put on the neck', especially the inexpensive, OTC liquids from the pet store, are recommended for use on pets, especially cats. Do not use the products labeled for dog use only, either intentionally or by accident. Mistakes happen, but be especially careful about this.
Fleas on newborns or pre-weaned kittens is an especially tough problem. Unchecked, they can kill a kitten, or a whole litter, from loss of blood. The safety of the nest and mother's care is killing them, because the flea population is concentrated in and around the kittens 'nest', and on Momma.
The female flea is a prodigious egg producer, producing up to 2,000 eggs per season, laying them mostly on the pet, where they eventually fall onto the ground, or rug, or wherever the pet travels or pauses throughout the day.
Fleas are primarily an environmental problem, as anyone who has, or had, a pet with an 'occasional' flea can testify, starting along in the Fall of the year. "Fleas seem to have just invaded the house, almost overnight!" Or, maybe you have gone on a two week late summer vacation and boarded the pet. The first night back home is miserable because the new fleas have been anxiously awaiting your return. The point here is that completely eliminating them is complicated by the fact that the only fleas you probably ever see are the ones on your pet, and occasionally on your socks or pant legs. But there are LOTS more than that around the house.
I wish there were an easy, simple way to eliminate a flea problem, but they have been a problem for man and animal for, maybe, ever, and they don't seem to be letting up. Chemicals, properly chosen and used, have made life easier for countless people and animals, and shorter for even more fleas. One of the more effective control measures involves adding a growth inhibitor to the flea control product... it stops the development of the egg, fairly rapidly reducing the population of reproducing fleas in the pet's environment. I keep saying 'environment'... a flea is an on/off/on/off the pet pest, and all the new fleas develop in the environment, not on the pet. The little bitty fleas are the 'new' ones. Just because there are two species of fleas, one dog and one cat, doesn't mean they are particular about who they bite. The cat species of flea is far and away the most common of the two and affects dogs as much as cats. Differences between the two species are best left to flea aficionados to see, as they are microscopic. Tiny fleas are very young ones. Big ones are probably females.
If your pet spends much time outdoors, and part of that in it's favorite shady spot under a porch, that is going to be a flea hot spot, and should be treated to eliminate the adults not on your pet, and the young as soon as they hatch. Interestingly, the flea develops in a cocoon like structure, like a butterfly. When it has matured, it partially opens the cocoon and then patiently waits till something comes close. The presence of vibrations from footsteps is enough stimulus for it to escape the cocoon and spring onto the new host... probably your pet. Or you, if this happens indoors. Indoors... in a dark and quiet closet, in the rug behind the sofa or chair, down the basement... anywhere your pet frequently rests, or hides, is candidate to be the indoor hot spot. Or spots, as there is no reason to not have several. Is there a favorite chair he or she likes to sit or sleep on? Your bed? When you're not home, maybe?
Now... how to clean all these places up? Well, there are chemicals, professionally applied. That isn't the entire answer either. You are supposedly going to do all this for you and your pet's sake. Your pet, especially the young kittens, probably can't afford to wait for everything, beside themselves, to be rid of fleas. A kitten will be killed in a couple weeks of heavy flea infestation from blood loss. We may have to take a few chances with it and use something 'toxic', especially to fleas.
Probably the safest thing to do for a kitten is to bathe it. I wouldn't waste your time with flea soaps. For the good they do, or don't, just stick with regular soaps. Residual flea effects are greatly diminished when you rinse it off, and you have to do that. Baby shampoo is quite gentle, and if/when it gets in the eyes, it isn't quite so bad. I think Dial dishwashing liquid would be harder on fleas however. Fleas don't like water and seem to keep ahead of it when you bathe the pet. Since we always start with the back end of the animal, the front end (ears, eyes, nose, mouth) winds up with all the fleas. Start with a good wetting and sudsing of the head and work backwards, before setting it in the water. Don't hurry. Maybe add the Dawn from the shoulders back, and wash clear to the end of the tail. Drain the sink (you aren't going to use the whole tub for a little kitten, are you?) and rinse the kitten fairly well. Use a fine tooth comb because you'll see several to quite a few fleas immobile in the wet hair. Comb 'em out, finish with a final, and thorough rinse, and down the drain they go. Dry kitty, and keep warm. You will probably have to repeat this several times until there is little evidence of fleas on the kids. As I said, the heavy infestation is what kills the babies, not an occasional flea.
Now, what are you going to do about Mom? Soon as she gets back in with the kids, they will be covered again, just not as bad, but soon she will have carried a bunch back to them. Check with your vet about whether Bob Martin will appear in the milk, or not. It does, after all, appear in the bloodstream. Some things can't really be helped. Let's hope Mom doesn't react to the treatment. If so, she gets a good bath ASAP if so.
A non chemical treatment that is inexpensive and relatively safe is diatomaceous earth. It's an extremely fine powdery substance made of ancient sea deposits of diatoms... microscopic organisms like, but much smaller than plankton. This is so fine it plugs up the spiracles, or 'breathing' holes in the flea, or any other insect's outer cuticle, and they smother. It is also so fine that it quite dusty and can easily be inhaled deep into the lungs. Not so 'fine'. So otherwise it may be good on animals. It is safe, otherwise... completely non toxic. You may try it on carpeted areas. Hardwood floors usually have a small space between the boards... good places for flea eggs to develop. Cement floors, like basements.. also plenty of holes for eggs. And also a very good candidate for a hot spot. If you find it impossible to get rid of the fleas, you have probably missed a hot spot, indoors or out if Mom goes outdoors.
If you're not real successful in getting most of the fleas off the kittens, you may have to resort to some chemical treatment (besides the Bob Martin approach). Sometime you will be wanting to show these guys off to the vet, right? At least, maybe speak to him about the problem and get his suggestion. I hope he's not one of those who won't talk to you.
A little long winded, but I hope there is some help in some of it. Best wishes.