My understanding of mullein is that it is a natural expectorant. It helps loosen things in the lungs so that the lungs can expel the mucus. If the mucus continues to sit in the lungs, it can lead to infection, which leads to the body trying to fight the infection, which leads to more mucus, and on and on it goes. Sorry - I don't have my books here right now, or I would look up and quote stuff for you.

They always seem to be able to explain things much better than I can.

My sister's son has what we are pretty sure is croup right now. I have a nebulizer, so made a really strong mullein tea for her to use in the nebulizer for him. She started it on him on Tuesday night and has been doing it 2-3 times a day for about 20 min. She said that he was so much better yesterday morning. Today she told me that he is better still. He coughs more whenever she uses it, but she said it does seem to be clearing out his lungs. He does not sound wheezy any more either. Each morning he has woken up less congested than before.
If you don't have a nebulizer, you can make the strong tea in a pot. Then, while it is still steaming, have them lean over it with a towel over their head to keep the steam in. Breathe the steam in deeply. I imagine you could have them breathe in over a vaporizer if you put the herb in the medicine cup like I mention down below.
I have used mullein as a tea with my kids whenever they are congested. It has always seemed to work great for clearing the lungs out. I usually use it in conjunction with camomile and orange peel. It is actually a great tasting tea with all three in there! If you sweeten it with some raw honey, that can help coat the throat too if it is irritated. I just let them drink that tea all day and usually we do not have too much of a battle getting over congestion this way. I also use elderberry syrup - but notice that you've used that also.
The other thing I've done with mullein is to wet the herb and put some in cheesecloth in the medicine cup of the vaporizer and let the vaporizer run all night. That seems to help.
Mullein is a great herb for kids, too. Lobelia is supposed to be a great herb for lung health, but it can be toxic if taken in large amounts. So I try to stick to mullein with my kids.