We consider the marginal problem in Dempster-Shafer theory, investigating the structure of a suitable set of bivariate joint belief functions having fixed marginals, by relying on copula theory. Next, we formulate two Kantorovich-like optimal transport problems, either seeking to minimize the Choquet integral of a given cost function with respect to the reference set of joint belief functions or its dual functional. We finally give a noticeable application by choosing a metric as cost function: this permits to define pessimistic and optimistic Choquet-Wasserstein pseudo-distances, that can be used to compare belief functions on the same space.