  Motivated by applications in model-free finance and quantitative risk
management, we consider Fr\'echet classes of multivariate distribution
functions where additional information on the joint distribution is assumed,
while uncertainty in the marginals is also possible. We derive optimal
transport duality results for these Fr\'echet classes that extend previous
results in the related literature. These proofs are based on representation
results for increasing convex functionals and the explicit computation of the
conjugates. We show that the dual transport problem admits an explicit solution
for the function $f=1_B$, where $B$ is a rectangular subset of $\mathbb R^d$,
and provide an intuitive geometric interpretation of this result. The improved
Fr\'echet--Hoeffding bounds provide ad-hoc upper bounds for these Fr\'echet
classes. We show that the improved Fr\'echet--Hoeffding bounds are pointwise
sharp for these classes in the presence of uncertainty in the marginals, while
a counterexample yields that they are not pointwise sharp in the absence of
uncertainty in the marginals, even in dimension 2. The latter result sheds new
light on the improved Fr\'echet--Hoeffding bounds, since Tankov [30] has showed
that, under certain conditions, these bounds are sharp in dimension 2.
