For two Markov chains:
$$X \rightarrow f(X)$$ $$Y \rightarrow f(Y)$$
where $f$ is 1-Lipschitz, if one minimizes the Wasserstein distance between $f(X)$ and $f(Y)$, does this result in the minimization of the Wasserstein distance between $X$ and $Y$?
The intuition is that this is correct since for a 1-Lipschitz function, the distance $D_A(f(X), f(Y)) \leq D_B(X, Y)$, i.e. the distance between $f(X)$ and $f(Y)$ is at most the distance between $X$ and $Y$ in another metric space, but I don't know how I could exactly prove this other than use some heuristic.
Any further intuition/ideas/proofs would be very helpful. Thank you so much.