It depends on the manufacturer. Some cheaper belts have a low quality, unwashed "core" material that may be pretty much any colour. Kwon, for example, has used brown-ish core material that, indeed, will dye the outer layer of belt brown when washed. Their coloured belts also loose colour rather heavily - the fabrics haven't been prewashed (or preshrunk, either) before assembly. Generally, if you pay $5-$10 or some other minimal sum, you should expect that your belt will loose colour in the wash. If you get a good quality colour belt (these exist, too - some are for competition purposes and some firms just have decided to put some effort to colour belts as well), you should expect to pay similar sums of money as you would for a good quality black belt.