  This paper is about the construction of displacement interpolations on a
discrete metric graph. Our approach is based on the approximation of any
optimal transport problem whose cost function is a distance on a discrete graph
by a sequence of Schr\"odinger problems associated with random walks whose jump
frequencies tend down to zero. Displacement interpolations are defined as the
limit of the time-marginal flows of the solutions to the Schr\"odinger
problems. This allows to work with these interpolations by doing stochastic
calculus on the approximating random walks which are regular objects, and then
to pass to the limit in a slowing down procedure. The main convergence results
are based on Gamma-convergence of entropy minimization problems. As a
by-product, we obtain new results about optimal transport on graphs.
