Suppose $\mu=\sum_{i=1}^k p_i\delta_{x_i}$ and $\mu'=\sum_{i=1}^k p_i'\delta_{x_i'}$ are two atomic probability measures. Let $W_r(\mu,\mu')$ be the Wasserstein distance between $\mu$ and $\mu'$. Are there any known bounds of the form $$ \sup_i\inf_j|p_i-p_j'| \le C\cdot W_r(\mu,\mu') $$

for some constant $C$ that may depend on quantities such as the number of atoms $k$? It is not hard to prove such a bound for the largest difference between the *atoms* $\sup_i\inf_jd(x_i,x_j')$, however, I am not sure if it is possible to derive something similar for the probabilities.

Another more concrete perspective: If $W_r(\mu_n,\mu')\to0$ and $\mu_n,\mu'$ all have exactly $k$ atoms, then it is a standard result that there are permutations $\pi_n$ such that $x_{\pi_n(i),n}\to x_i'$ and $p_{\pi(i),n}\to p_i'$. Now suppose $W_r(\mu_n,\mu')\le r_n$ for some sequence $r_n>0$. Does it follow that the rate of convergence of the probabilities is also $r_n$? This is easy to show for the atoms, but I am not sure about the probabilities.