You might enjoy arxiv.org/abs/1412.4430. It has a nice duality-based derivation of the dynamic approach to optimal transport (together with some sweet reasoning for why the drift is of gradient type) which can potentially be translated into your discrete-time setting to match …
Who's Who of AI
Sam Power
Lecturer in Maths & Stats at Bristol. Interested in probabilistic + numerical computation, statistical modelling + inference. (he / him).
Homepage: https://sites.google.com/view/sp-monte-carlo
Seminar: https://sites.google.com/view/monte-carlo-semina
What they're sharing
arxiv.org