For a problem I've been working on, I'm thinking about couplings between true and empirical distrubutions.

I have two datasets $S$ and $T$ with underlying measures $\mu_S,\,\mu_T$. And then I have some coupling on those, say $\pi\in \Pi(\mu_S,\mu_T)$ with $\mu_S$ and $\mu_T$ as marginals.

Now if I come up with empirical distributions $\hat{\mu}_S$ and $\hat{\mu}_T$ based off of sample draws from $S$, $T$, is there anything I can say about the couplings $\hat{\pi} \in \Pi(\hat{\mu}_S,\hat{\mu}_T)$?

Or at least any literature out there that discusses this?