  Let $M$ be a closed Riemannian manifold with a family of Riemannian metrics
$g_{ij}(t)$ evolving by a geometric flow $\partial_{t}g_{ij} = -2{S}_{ij}$,
where $S_{ij}(t)$ is a family of smooth symmetric two-tensors. We derive
several differential Harnack estimates for positive solutions to the nonlinear
backward heat-type equation \begin{eqnarray*} \frac{\partial f}{\partial t} =
-{\Delta}f + \gamma f\log f +aSf \end{eqnarray*} where $a$ and $\gamma$ are
constants and $S=g^{ij}S_{ij}$ is the trace of $S_{ij}$. Our abstract
formulation provides a unified framework for some known results proved by
various authors, and moreover lead to new Harnack inequalities for a variety of
geometric flows.
