Hi Muttley. If you have the atopy gene and therefore a predisposition to eczema and asthma - it's a very good idea to eliminate dairy from your diet.
My first two children suffered badly with eczema so when #3 came along unexpectedly 10 years later we started him on a dairy-free diet from birth. Thus far no eczema or asthma (which the other two developed at 7). We soon know if any illicit milk chocolate, etc has been consumed with raw patches breaking out on his face, usually behind his ears.
Out of support for #3 I decided to give up dairy at the same time and found it has not only helped my bad skin but resolved some gut issues and improved breathing! (Casein in cow's milk stimulates mucus production.) I still have asthma but perhaps it's not as bad as it would have been?
I'm not absolutely strict about it - I find I can eat hard cheese with no adverse effect, and it is very hard to avoid dairy when you're eating out, especially if you're a vegetarian as I am. There isn't much left on the menu!!
As Caramel M says some folks do have gut problems with soya so don't go overboard on it. There are non-soya alternatives to milk and cheese.
I think you'll need to exclude all dairy for about 2 weeks before you notice the benefits. Let us know how you get on.