I am not a herbalist. Try the horizon website. I think they have really the best choice of herbs and there is a description with each herb.
I read somewhere that coffee helps for asthma, well it doesn't really help only alleviate the symptoms. Yoga helps.
Maybe you try bitter vegetables. Dandelion in spring or old fahioned varieties of Brussels Sprouts or bitter melon. It is true that many vegetables had more bitter flavour, but now they brought up new varieties without bitter flavour. Many wild vegetable herbs are a bit bitter.
The Greek make horta (I have no idea how to spell this correctly). They take green stuff from the market but often you see them searching for wild greens.
You simply boil the greens briefly maybe 3-5 minutes in salt water, drain and pour good green olive oil over it and lemon juice. And it is mostly eaten with fish.
But this is an adult only vegetable, kids will find it yuck.
We are in a prime asthma region. There's pollen year round. And people say that you have to eat local honey this is like a vaccination against asthma.
Swimming is good and yoga.
My husband has asthma and he does not take medications. If it's bad he goes to a Chinese practitioner.