There are several asthma inhalers that come in a dry powder form. The most common are Asmanex, Advair, and Foradil, although there may be others.
A dry powder inhaler may deposit powder on your vocal cords, causing hoarseness and coughing. This can happen even if you are very good about rinsing your mouth after using your inhaler.
If you are taking a dry powder inhaler and experience hoarseness and/or coughing, discuss it with your asthma doctor and ask whether you could try a similar type of inhaler that is not dry powder. The link above from one of the professional allergists' associations lists different types of asthma inhalers.
Also, it's important to know that some of these dry powder inhalers (e.g., Asmanex, Foradil) contain lactose, a milk product, so if you or your child are allergic to milk, you should avoid those.