An increase in mucus can happen with asthma, but not everyone with asthma has mucus. How much mucus is very different from 1 person to another. Smoking definitely increases mucus.
In some people dairy products coat the mucus making it feel thicker and sticker. It is not known if this is caused by the fat in the dairy product or something else. Drinking plenty of water will help to thin the mucus so that it moves more easily.
It is unusual for foods to trigger asthma. Some people can be allergic to foods like peanuts, nuts, and shellfish. When eaten by an allergic person the food causes a really bad reaction with breathing difficulty. This is called anaphylaxis. It is not asthma.
I have asthma, all milk products have a problem with me. Sometimes eggs does too. Though I have been going to the organic store for my shopping. I've notice some soy products produce mucus also. I'm allergic to shell fish and smells of smoke which can trigger asthma. Then I cough in order to breathe, that forms the buildup of mucus in the lungs.