  We prove that the sequence of marginals obtained from the iterations of the
Sinkhorn algorithm or the iterative proportional fitting procedure (IPFP) on
joint densities, converges to an absolutely continuous curve on the
$2$-Wasserstein space, as the regularization parameter $\varepsilon$ goes to
zero and the number of iterations is scaled as $1/\varepsilon$ (and other
technical assumptions). This limit, which we call the Sinkhorn flow, is an
example of a Wasserstein mirror gradient flow, a concept we introduce here
inspired by the well-known Euclidean mirror gradient flows. In the case of
Sinkhorn, the gradient is that of the relative entropy functional with respect
to one of the marginals and the mirror is half of the squared Wasserstein
distance functional from the other marginal. Interestingly, the norm of the
velocity field of this flow can be interpreted as the metric derivative with
respect to the linearized optimal transport (LOT) distance. An equivalent
description of this flow is provided by the parabolic Monge-Amp\`{e}re PDE
whose connection to the Sinkhorn algorithm was noticed by Berman (2020). We
derive conditions for exponential convergence for this limiting flow. We also
construct a Mckean-Vlasov diffusion whose marginal distributions follow the
Sinkhorn flow.
