Thyme
Thyme is an excellent lung herb and is particularly useful for coughs.
It used to be prescribed for whooping cough in children.
Its main mode of action is through the volatile oil. Even when given by mouth the oil is eliminated largely via the lungs so that it is concentrated exactly at the site where its effect is required.
If you use the fresh herb a good handful mixed into the feed twice a day should be sufficient.
This is not a herb that you need to use all the time; just if your horse has a stubborn cough.