There is a close relationship between Optimal Transport and Field Alignment methods. For example, Haker et al. (IJCV 2004) use it for warping, and it was used earlier in EMD in Computer Vision. See also Rachev and Ruschendorf, 1998 (https://www.springer.com/la/book/9780387983523) for several others. 

Field Alignment methods embed a form of Optimal Transport. While the Kantorovich–Wasserstein metric  provides a formulation for Field Alignment, it is closely related also to Benamou and Bernier's work. While the original form is related to diffeomorphic approaches but subsequent ones (e.g., scale cascaded alignment) differ. When Optimal Transport is defined as being divergence-free deformation between the source and destination mass distributions, then correspondence with nondivergent solutions of Field Alignment emerge.  However, our approach considers Field Alignment in the context of a Bayesian coupling between Optimal Transport and Optimal Estimation. This is substantially more general in the sense that the difficulty in identifying appropriate densities while maintaining their mass conservation in the presence of noisy mesurements, and regularizing the correspondence map between the domains when one domain is very sparsely observed remain as issues  As Haker et al. note, it's correspondence free, symmetric and can work across changes in "density" (intensity) -- however, there are many implementation issues for this to work to still be considered.

Despite the recent interest in Optimal Transport in Data Assimilation, we posit that extant methods (e.g. Ravela et al. 2007) have previously considered flows (transport) of fields in a variational setting for data assimilation.  Furthermore, we think that the Bayes-Coupled Optimal Transport and Estimation (B-COTE) offers a solution the conundrum of whether amplitude adjustment or geometric adjustment is to be preffered and how. 

Thus, while deformation approaches developed thus far including: FABle (Field Alignment Blending), AROMA (Adaptive Reduced Order Modeling By Alignment), Field Coalescence, Coherent Random Fields and the two variants of Field Alignment can be cast within Wasserstein, it is important to note that the variational formalism (including ensemble solutions) provide a key generalization.