  Large optimal transport problems can be approached via domain decomposition,
i.e. by iteratively solving small partial problems independently and in
parallel. Convergence to the global minimizers under suitable assumptions has
been shown in the unregularized and entropy regularized setting and its
computational efficiency has been demonstrated experimentally. An accurate
theoretical understanding of its convergence speed in geometric settings is
still lacking. In this article we work towards such an understanding by
deriving, via $\Gamma$-convergence, an asymptotic description of the algorithm
in the limit of infinitely fine partition cells. The limit trajectory of
couplings is described by a continuity equation on the product space where the
momentum is purely horizontal and driven by the gradient of the cost function.
Convergence hinges on a regularity assumption that we investigate in detail.
Global optimality of the limit trajectories remains an interesting open
problem, even when global optimality is established at finite scales. Our
result provides insights about the efficiency of the domain decomposition
algorithm at finite resolutions and in combination with coarse-to-fine schemes.
