Abstract: The discrete case is described in the recent works [1] and [2], the continuous case is the subject of the ongoing research.
[1] Backhoff, Bartl, Beiglböck, Eder. All adapted topologies are equal. Probab. Theory Relat. Fields 178, 1125–1172 (2020). https://doi.org/10.1007/s00440-020-00993-8
It can be found here: https://link.springer.com/article/10.1007/s00440-020-00993-8
[2] Bartl, Beiglböck, Pammer. The Wasserstein space of stochastic processes.
It can be found here: https://arxiv.org/abs/2104.14245