Let $M$ be a smooth manifold, let $I \subset \mathbb{R}$ be an interval, and let $g:I\to\Gamma(\operatorname{Sym}^2T^*M)$ be a smooth one-parameter family of Riemannian metrics satisfying $\partial_t g(t)=-2\operatorname{Ric}(g(t))$ for every $t\in I$. If $\phi:M\to M$ is a fixed diffeomorphism, then $\tilde g:I\to\Gamma(\operatorname{Sym}^2T^*M)$, defined by $\tilde g(t)=\phi^*g(t)$, is also a Ricci flow.