I've come across various brining solutions that include sugars: white sugar, brown sugar, raw sugar, etc. Are these sugars interchangeable? Is one sugar more desirable for a certain kind of meat over the other?
For a brine, no it doesn't really matter, except that granulated sugar will dissolve more nicely than raw sugar. The sugar is for flavor more than anything else, though it does help the brine some. It also promotes browning of the meat. If you change one kind of sugar for another, I'd use a 1:1 swap by weight rather than volume so you don't have to worry about crystal sizes.
Though the sugar does help the brining action some, the salt is the important component.
From Cook's Illustrated:
How does brining work? Brining promotes a change in the structure of the proteins in the muscle. The salt causes protein strands to become denatured, or unwound. This is the same process that occurs when proteins are exposed to heat, acid, or alcohol. When protein strands unwind, they get tangled up with one another, forming a matrix that traps water. Salt is commonly used to give processed meats a better texture. For example, hot dogs made without salt would be limp.
In most cases, we add sugar to the brine. Sugar has little if any effect on the texture of the meat, but it does add flavor and promotes better browning of the skin.