Bronchitis. is it possible for my mother to pass her asthmatic bronchitis on to her grandson?

Yes: If asthma is what you mean by asthmatic bronchitis. Asthma like allergic rhinitis & eczema has a strong genetic component. It's common to see several generations of patients from 1 family. But perhaps you mean that when your mother smokes she gets bronchitis & now your son is coughing. You can't change genetics but you can environment. 2nd-hand smoke is a reason why grandma & grandson are coughing.

Yes.: Asthma tends to be more common if one or both parents have an allergic problem such as hay fever or asthma.