Suppose on a manifold $M$ I have a family of complete vector fields, $V(t) \in \Gamma(TM)$ for each $t \in \mathbb R$ with $V(0) = 0$. Also assume that $[V(t_0), V(t_1)]=0$ for any $t_0, t_1 \in \mathbb R$. Let $s \rightarrow \phi_s^X(p)$ be the integral curve of the vector field $X$ with initial condition $p \in M$, parametrized by $s$. My question is, is there any way to compute $$ \frac{d}{dt} \phi_s^{V(t)}(p) $$ for given fixed $s \neq 0$ and $p \in M$? For example, I am trying to compute $$ \frac{d}{dt}\rvert_{t=0} \phi_1^{V(t)}(p) $$ for such a family.
I tried looking at the theorem for flows of time-dependent vector fields to obtain some chance of computation. There, an integral curve of $V$ with initial condition $(t_0, p) \in \mathbb R \times M$ is a curve $\psi^{(t_0,p)}: \mathbb R \rightarrow M$ such that $$ \frac{d}{dt} \psi^{(t_0,p)}(t) = V(t)_{\psi^{(t_0,p)}(t)} $$
According to Theorem 9.48 of Lee's "Introduction to smooth manifolds", such integral curves always exist and have good composition properties.
However, I failed to see any relationship between these integral curves and the flows $\phi_t^{V(t)}(p)$. Initially I had thought it reasonable that $$ \psi^{(t_0,p)}(t) = \phi_{t-t_0}^{V(t)}(p) $$ but I don't actually know how to prove that this is the case, and I feel like what I'm asking is equivalent to proving that this is the case; both computations seem to reduce to the question of how to differentiate the flow respect to $t$ when $V(t)$ changes.