Closed-form analytical solutions to Optimal Transport/Wasserstein distance

Although a bit old, this is indeed a good question. Here is my bit on the matter:

Regarding Gaussian Mixture Models: A Wasserstein-type distance in the space of Gaussian Mixture Models, Julie Delon and Agnes Desolneux, https://arxiv.org/pdf/1907.05254.pdf
Using the 2-Wasserstein metric, Mallasto and Feragen geometrize the space of Gaussian processes with $L_2$ mean and covariance functions over compact index spaces: Learning from uncertain curves: The 2-Wasserstein metric for Gaussian processes, Anton Mallasto, Aasa Feragen https://papers.nips.cc/paper/7149-learning-from-uncertain-curves-the-2-wasserstein-metric-for-gaussian-processes.pdf
Wasserstein space of elliptical distributions are characterized by Muzellec and Cuturi. Authors show that for elliptical probability distributions, Wasserstein distance can be computed via a simple Riemannian descent procedure: Generalizing Point Embeddings using the Wasserstein Space of Elliptical Distributions, Boris Muzellec and Marco Cuturi https://arxiv.org/pdf/1805.07594.pdf (Not closed form)
Tree metrics as ground metrics yield negative definite OT metrics that can be computed in a closed form. Sliced-Wasserstein distance is then a particular (special) case (the tree is a chain): Tree-Sliced Variants of Wasserstein Distances, Tam Le, Makoto Yamada, Kenji Fukumizu, Marco Cuturi https://arxiv.org/pdf/1902.00342.pdf
Sinkhorn distances/divergences (Cuturi, 2013) are now treated as new forms of distances (e.g. not approximations to $\mathcal{W}_2^2$) (Genevay et al, 2019). Recently, this entropy regularized optimal transport distance is found to admit a closed form for Gaussian measures: Janati et al (2020). This fascinating finding also extends to the unbalanced case.