Mapping clusters between datasets

I am using MVN mixture model clustering to cluster observations in data with D features into K classes. I fit my algorithm to few different data sets, with a key assumption that the clusters are consistent across the datasets except for some amount of drift in the cluster means. In this unsupervised case, the cluster labels don't automatically match between the data sets. How can I make them match?

I thought of doing a stochastic search of the space of orderings of the elements of the mean vectors in each dataset, seeking the set of orderings that minimizes Euclidean distance between each set of mean vectors. But that seems like overkill. I'm certain someone has solved this by now, anyone know how?