We study the distributionally robust optimization (DRO) in a dynamic context where the model uncertainty is captured by penalizing potential models in function of their adapted Wasserstein distance to a given reference model. We consider both discrete- and continuous-time settings and derive dynamic duality formulas that reformulate the worst-case expectation as a tractable minimax problem. The inner maximum can be computed recursively in discrete time, or solved by a path-dependent Hamilton--Jacobi--Bellman equation in continuous time. We further extend these duality results from the worst-case expectation to the worst-case expected shortfall, a non-linear expectation. Finally, we apply the DRO framework to optimal stopping problems in discrete time. We recast the original problem as a classical Wasserstein DRO on a nested space by introducing a novel relaxation that considers stopping times with respect to general flitrations.