The tires only turn brown in some cases because the dust/dirt, etc is attracted to freshly-dressed tires. The reason for this? Most tire dressings (except water-based ones.. which do not last very long.. 1 day tops) contain silicone which attract dust, dirt, etc while driving. When washing your vehicle you can usually avoid this by spraying some tire cleaner onto the tires which should remove the build-up.
In the end its a give and a take. On one hand a freshly-cleaned/detailed vehicle really stands out once the tires are dressed (which also helps protect the rubber from cracking and harmful UV rays) however, on the other hand it opens the door for tires attracting dust/dirt, etc and sometimes turning brown or dull. Personally, I find that dressing the tires on a vehicle is what makes the entire cleaning/detailing worth it, as it accents the nice shiny paint
