This is remark 1.24 on p. 13 of the book Hamilton's Ricci Flow by Bennett Chow, but how to prove this conclusion?
If $\varphi (t): M^n \to M^n$ is the $1$-parameter family of diffeomorphism and $\alpha$ is a tensor, then $ \frac{\partial}{\partial t} (\varphi(t)^\ast \alpha) = L_{X(t)} \varphi(t)^\ast \alpha ,$ where $ X(t_0) = \left. \frac{\partial}{\partial t} \right|_{t = t_0} (\varphi (t_0)^{-1} \circ \varphi(t)) . $