Many evolutionary partial differential equations may be rewritten as the gradient flow of an energy functional, a perspective which can provide useful estimates on the behavior of solutions. The notion of gradient flow requires both the specification of an energy functional and a metric with respect to which the gradient is taken. In particular, much recent work has considered gradient flow in the Wasserstein metric. Given the formal nature of the gradient in this setting, a useful technique for constructing solutions to the gradient flow and studying their stability is to consider the "discrete gradient flow", a time discretization of the gradient flow problem analogous to the implicit Euler method in Euclidean space. In this talk, I will present a new proof that the discrete gradient flow converges to the continuous time gradient flow inspired by Crandall and Liggett's result in the Banach space case. Along the way, I will discuss theorems of independent interest concerning transport metrics.