Uniqueness of the limit of a set of probability measures (discrete)

Let $ K subset mathbb {R} ^ n $ be compact and consider a sequence of probability measures $ mu_n $ sure K $. Being weakly bounded, we know that there is a weakly convergent subsequence of the $ mu_n $. By weak convergence, I mean the convergence given by the action against $ C (K) $. My question is what kind of conditions can be imposed on the sequence $ mu_n $ to ensure that the original sequence converges as opposed to a subsequence?

In relation, let me change the question by asking now that the $ mu_n $ to be discreet that is to say $ mu_n = frac {1} {n} sum_ {i = 1} ^ n delta_ {x_i} $ for a set (possibly according to n) of points $ x_1, …, x_n in K $.