Expressing a curve reparametrization as a diffeomorphism

Question

I have some dynamics given on a manifold $\tilde{N}$ through a vectorfield $\tilde{X}\in\Gamma(T\tilde{N})$ on $\tilde{N}$, namely the dynamics are the flow of $\tilde{X}$. So for instance if $c(t)$ is a curve of my dynamics then \begin{equation} \dot{c}_{c(t)}=\tilde{X}_{c(t)}. \end{equation} Furthermore I have a really important reparametrization $s\mapsto t$. Now I am working on something that requires me to work with vectorfields and flows directly (no reparametrization). I am wondering if I can somehow map the manifold (to itself or a similar one $N$) with a diffeomorphism $F$ such that the push forward vectorfield $dF\cdot\tilde{X}$ is such that if I build its flow it yields the reparametrized dynamics, i.e. a curve $c(s)$ of its dynamics satisfy \begin{equation} \frac{d}{ds}c(s(t))_{c(s(t))} = dF_{c(t)}\cdot \tilde{X}_{c(t)}. \end{equation}