What really helped me with my very mild asthma was learning that asthma doesn't keep me from breathing air in, it keeps me from forcing air out. Since my asthma is so mild, I'd never seen a pulmonologist, so I learned this from a former riding buddy who has EIA.

Forget trying to inhale. The reason it feels like you can't get air in is because your lungs are already full (of spent air). Use those assistive muscles to forcibly exhale as much as you can. It's amazing how much better I breathe once I learned this.