You might look at Chapter 2 of my book Lipschitz Algebras. The Banach space ${\rm Lip_0}(X)$ is already the dual of the space of finitely supported measures on $X$ satisfying $\mu(X) = 0$, equipped with Wasserstein distance (though I suppose it should then be called Arens-Eells distance). Going to Radon measures enlarges this space but you remain within the completion of the finitely supported measures, so its dual space doesn't change.

You might look at Chapter 3 of my book Lipschitz Algebras (second edition). The Banach space ${\rm Lip_0}(X)$ is already the dual of the space of finitely supported measures on $X$ satisfying $\mu(X) = 0$, equipped with Wasserstein distance (though I suppose it should then be called Arens-Eells distance). Going to Radon measures enlarges this space but you remain within the completion of the finitely supported measures, so its dual space doesn't change.