Gaussian Mixture Models (GMMs) are ubiquitous in statistics and machine learning and are especially useful in applied fields to represent probability distributions of real datasets. Optimal transport can be used to compute distances or geodesics between such mixture models, but the corresponding Wasserstein geodesics do not preserve the property of being a GMM. In this talk, we show that restricting the set of possible coupling measures to GMMs transforms the original infinitely dimensional optimal transport problem into a finite dimensional problem with a simple discrete formulation, well suited to applications where a clustering structure is present in the data. We also present possible extensions of this Wasserstein-type distance between GMMs that remain invariant to isometries. Inspired by the Gromov-Wasserstein distance, these extensions can also be used to compare GMMs of different dimensions.