  In this note we will adapt Topping's $\mathcal{L}$-optimal transportation
theory for Ricci flow to a more general situation, i.e. to a closed manifold
$(M,g_{ij}(t))$ evolving by $\partial_tg_{ij}=-2S_{ij}$, where $S_{ij}$ is a
symmetric tensor field of (2,0)-type on $M$. We extend some recent results of
Topping, Lott and Brendle, generalize the monotonicity of List's (and hence
also of Perelman's) $\mathcal{W}$-entropy, and recover the monotonicity of
M$\ddot{u}$ller's (and hence also of Perelman's) reduced volume.
