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Certainly there are non-uniqueness issues. I actually meant something like: given a Wasserstein geodesic $\mu_t$, can we produce sequences $(\mu_{0,n})$ and $(\mu_{1,n})$ such that $(\mu_{t,n})$ converges to $\mu_t$ in some suitable sense.
Alfred Galichon has done some work along these lines, for matching problems in an aggregate utility setup. Have a look at his book, or the paper "Personality traits and the marriage market" by Dupuy and Galichon.