Yes, they do.
Humidity and sweat will cause the belt to shrink a little bit over time. It's what we call in Japan "the belt is alive". We've also had a few rares cases when the belt shrank during transportation from Japan to the customer's address overseas.
If you are going to wash the belt then it will shrink. We do not recommend washing the belt, but some people like to do it. Please keep in mind that even if you do not wash the belt, it will naturally shrink from sweat and moist.
In our black belt pages you'll find our estimated shrinkage rates for each belt model. For example: