1- It is not usual for a kitten to die suddenly from intestinal worms. Normally worms result in a slow decline with the kitten becoming less active, lethargic, losing interest in eating. If the worms cause an intestinal blockage, the kitten would be vomiting and in severe abdominal pain.
2- The most common intestinal worm in kittens by far is the roundworm. Children can get them via poor hygiene. A common scenario would be for the child to play in an unclearned cat box or in a sandbox where the cat has been pooping, then putting their fingers into their mouth. Another route, though less common, would be where the cat is routinely using the garden for a toilet and the child is picking and eating raw, unwashed vegetables in contact with the soil, such as radishes, carrots, etc.
Roundworms can be serious for some children, though most children only have mild symptoms. But in rare cases they can result in blindness, intestinal blockage, severe allergic reactions, and more. I am not a human physician, but my suggestion would be to have your children tested if you fear that they may have been exposed. The test is simple, a case of submitting a sample of the feces to a lab.
In my 30+ years of veterinary experience, I have never seen
DE work sucessfully as a deworming method, though I have had many, many clients try it on their pets. I have no experience with its use in humans.
...Su Ba