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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.