  In this paper we prove first order differential Harnack estimates for
positive solutions of the heat equation (in the sense of distributions) under
closed Finsler-Ricci flows. We assume mild non-linearities (in terms of the
Chern connection, $\mathbf{S}-$curvature and Hessian) and suitable Ricci
curvature bounds throughout the flow. One of the key tools we use is the
Bochner identity for Finsler structures proved by Ohta and Sturm.
