I've seen the following equality being used in a book on the Ricci flow I'm reading but I've been stuck on proving it:
$$\varphi_{t}^{*}\left(\mathcal{L}_{W(t)} g(t)\right)=\mathcal{L}_{\left[\left(\varphi_{t}^{-1}\right)_{*} W(t)\right]}\left(\varphi_{t}^{*} g(t)\right)$$
The context here is as follows: $(\mathcal{M}, g_0)$ is a Riemannian manifold on which $W$ is a time dependent vector field, $\{g(t) \}_{t \in I}$ is a family of time-dependent metrics on $\mathcal{M}$, and the $1$-parameter family of diffeomorphisms $\varphi_{t}: \mathcal{M}^{n} \to \mathcal{M}^{n}$ satisfies the non autonomous ODE:
$$\begin{aligned} \frac{\partial}{\partial t} \varphi_{t}(p) &=-W\left(\varphi_{t}(p), t\right) \\ \varphi_{0} &=\mathrm{Id}_{\mathcal{M}^{n}} \end{aligned}$$
I don't have a lot of experience with Lie derivative computations and this is the last step I need to finally finish a proof that solutions to the Ricci-deTurck flow can be produced by solutions from the Ricci-Hamilton flow. I know that if $M$ and $N$ are smooth manifolds, $F: M \to N$ is a diffeomorphism and $X \in \mathcal{X}(M)$ it's true that $$F_{*}\left(\mathcal{L}_{V} X\right)=\mathcal{L}_{(F_{*} V)} (F_{*} X)$$ but I'm having a hard time adapting this to pullbacks where everything is time dependent. This has been troubling me for some time now, so I'll be grateful for any help.
I don't think that anything in the equation depends on time varying. Or rather the equation is a relation at fixed arbitrary times $t=t_0$. So you just obtain a diffeomorphism $\varphi_{t_0}$ and vector field $W(t_0)$ as for your example with $F$ and $V$. The equation should be interpreted as a family of fixed time equations.