I want to prove the following (if it is true)
Let be $M$ a manifold, $\Lambda \in \Omega^k(M)$ a $k$-form on $M$, $X_t \in \mathfrak{X}(M)$ a time-dependent vector field on $M$ and $\phi_t \in Diff(M)$ a time-dependent diffeomorphism of $M$ such that $\phi_t' = X_t\phi_t$. Then for every $t$: $\mathcal{L}_{X_t}\Lambda = \frac{\partial}{\partial s}(\phi_{t+s}^{\ }\phi_t^{-1})^*\Lambda|_{s=0}$
My attempt was:
Consider the projection $\pi: \mathbb{R} \times M \to M$. Define $\bar{\Lambda} = \pi^*\Lambda \in \Omega^k(\mathbb{R} \times M)$. Define $\bar{X} \in \mathfrak{X}(\mathbb{R} \times M)$ such that $\bar{X}(t, p) = \partial_t + X_t(p)$. Define $\bar{\phi_s} \in Diff(\mathbb{R} \times M)$ such that $\bar{\phi_s}(t, p) = (t + s, \phi^{\ }_{t+s}\phi_t^{-1}(p))$. Then $\bar{\phi_s}$ is a flow for $\bar{X}$. So
$\frac{\partial}{\partial s}(\phi_{t+s}^{\ }\phi_t^{-1})^*\Lambda|_{s=0} = \frac{\partial}{\partial s}(\pi\bar{\phi_s}\pi_t^{-1})^*\Lambda|_{s=0} = {(\pi_t^{-1})}^*(\frac{\partial}{\partial s}\bar{\phi_s}^*\bar\Lambda)|_{s=0} = {(\pi_t^{-1})}^*\mathcal{L}_{\bar{X}}\bar{\Lambda}$
where $\pi_t = \pi|_{\{t\}\times M}$. What I can't prove (at least formally, because it seems obvious to me) is that ${(\pi_t^{-1})}^*\mathcal{L}_{\bar{X}}\bar{\Lambda} = \mathcal{L}_{X_t}\Lambda$ for every $t \in \mathbb{R}$.
EDIT
One idea is to use Cartan's formula and then discard the $dt$ term, but I hope there is a more elegant way.
Let $X_t$ be a time-dependent vector field on the manifold $M$ and let $\phi_t$ be its isotopy, i.e. $\phi_t$ is a family of diffeomorphisms of $M$ such that $\phi_0=\mathrm{id}_M$ and $$\forall p\in M\quad \partial_t\phi_t(p)=X_{\phi_t(p)}.$$ Then, by Proposition $6.4$ in da Silva's Lectures on Symplectic Geometry for any tensor $T$ you have $$\partial_t\ \phi_t^*T=\phi_t^*\left(\mathcal{L}_{X_t}T\right).$$ Now, rewrite the left-hand side as $$\partial_t\ \phi_t^*T=\partial_s\Bigr|_{s=0}\phi_{t+s}^*T$$ and you get $$\partial_s\Bigr|_{s=0}\phi_t^{-1}{}^*\phi_{t+s}^*T=\mathcal{L}_{X_t}T$$ that is, $$\partial_s\Bigr|_{s=0}\left(\phi_{t+s}\circ\phi_t^{-1}\right)^*T=\mathcal{L}_{X_t}T.$$