  We propose a scalable, distributed algorithm for the optimal transport of
large-scale multi-agent systems. We formulate the problem as one of steering
the collective towards a target probability measure while minimizing the total
cost of transport, with the additional constraint of distributed
implementation. Using optimal transport theory, we realize the solution as an
iterative transport based on a proximal descent scheme. At each stage of the
transport, the agents implement an online, distributed primal-dual algorithm to
obtain local estimates of the Kantorovich potential for optimal transport from
the current distribution of the collective to the target distribution. Using
these estimates as their local objective functions, the agents then implement
the transport by proximal descent. This two-step process is carried out
recursively by the agents to converge asymptotically to the target
distribution. We rigorously establish the underlying theoretical framework for
the algorithm and test its behavior via numerical experiments.
