The Wasserstein distance between mixing measures has come to occupy a central place in the statistical analysis of mixture models. We give the first axiomatic justification of its usage as a canonical measure of discrepancy between any mixture distributions. Inference for the Wasserstein distance between mixing measures is generally difficult in high dimensions. Specializing to discrete mixtures arising from topic models, we offer the first minimax lower bound on estimating the distance between pairs of mixing measures in this model class. This reveals regimes under which fast estimation of the distance between mixing measures can be expected, even if the ambient dimension of the mixture distributions is large. In such regimes, we develop fully data-driven inferential tools that allow us to obtain the first asymptotically valid confidence intervals for the Wasserstein distance between mixing measures, in topic models. Our results apply to potentially sparse mixtures of potentially sparse high-dimensional discrete probability distributions, and are illustrated via an example on movie reviews from the IMDb data set.