Different patinas can be caused by almost anything, including soils, moisture, and the coin itself.
In the case of "brown
patina", be very, very careful. that is
toning that is probably just above the bronze -- the substances that form first after a coin is minted, such as copper sulfides, are brown or black and it is only later that the green stuff starts to form. If you are hitting brown, you are very, very close to the metal of the coin.
My personal experience is that when coins have such varied surfaces, mechanical cleaning is the only way to go. Otherwise, any chemical treatment, and DW is a chemical treatment, affects all the parts of the coin equally, which is not what you want -- you want to focus on specific areas, I would think.
Mark