short answer is yes.
asthma is like an allergic reaction. certain things trigger the airways to narrow causing wheezing and in severe cases inability to breath. people with asthma are "atopic", it means they are more likely to have allergies, hayfever and eczema than the general population.
some asthmatics suffer in cold weather, some have problems with house dust, or pet hair. cigarette smoke is a classic allergen. exposure to it can trigger an asthmatic reaction. parents of asthmatic children are advised to never smoke indoors and to refrain from smoking in the presence of the child in question, even outdoors. ideally of course they should give up altogether.
so, residue can cause an asthmatic reaction. it depends on how sensitive the child is, some asthmatics have very mild asthma and wouldn't be bothered by smoke/dust/whatever as much as others. with children, particularly infants, its best not to take any risks, as an attack can be particularly dangerous for young children.
Medical school, asthmatic brother