Here's a couple feline asthma groups.
Post over there.
Realistically, I don't know there is *that*
much available info for asthmatic kittens.
They are definitely in the minority.
Only 1% of cats (of any age) get asthma.
Should have an xray and see what this kit's
lungs look like.
Asthma is often controlled with prednisolone
treatment, which isn't expensive at all.
A couple weeks of an inexpensive drug could
put the condition in remission for 6-12 months.
More serious cases will have to use inhaled meds,
and that is expensive. With asthmatic cats, there
commonly are triggers, and they differ with the cats.
For some, it's pollen, others dust, or smoke.
Figuring out what a cat is sensitive to, and
combatting the problem is obviously helpful.
Hepa filters in the house for example.

I'd have your normal vet take a look at the kitten.
Again it is unusual to have an asthmatic kitten,
and there are other conditions it could be. Some
shelters unfortunately aren't that reliable in
diagnosing problems. It could be something respiratory,
and this particular kitten could just need more than
the average regimen of antibiotics. Could be heart worm,
lung worm, a heart condition, asthma, or a few other things.

Feline asthma groups (some of the people there will know more):

http://groups.yahoo.com/group/felineasthma_inhaledmeds/
http://groups.yahoo.com/group/felineasthma/messages