How do functional language gui bindings work?

Question

Do they typically use non functional aspects of the language (including mutable variables). Are there other strategies? Could you describe them?

Answer 1

Do they typically use non functional aspects of the language (including mutable variables)

Many do, yes. In case of haskell, which doesn't have mutable variables or functions with side-effects, they use the IO monad (which is haskell's way of encapsulating side-effects)-

Are there other strategies? Could you describe them?

Yes, there are. Look at functional reactive programming.