I am a doctor & I am wondering that if something that passes through the throat & settles in lungs nullifies fast, than what about the air we breath, or the smoke & dust we inhale all the time? I request you to consider this aspect before declaring Inhaler for asthma as non permissible in fasting. In my humble opinion Inhaler is in vapour form, it does not provide any material for ingestion of that goes to the stomach, therefore it may be allowed. Further more allowing use of inhaler will allow many asthma patients to fast which they will be unable to do without the inhaler.
May Allah guide us all to the right path