  We study the problem of transporting one probability measure to another via
an autonomous velocity field. We rely on tools from the theory of optimal
transport. In one space-dimension, we solve a linear homogeneous functional
equation to construct a suitable autonomous vector field that realizes the
(unique) monotone transport map as the time-$1$ map of its flow. Generically,
this vector field can be chosen to be Lipschitz continuous. We then use
Sudakov's disintegration approach to deal with the multi-dimensional case by
reducing it to a family of one-dimensional problems.
