Suppose that $\Phi$ is the solution of $$\begin{cases} \frac{d}{dt}\Phi(x,t) = f(\Phi(x,t),t) \quad t >0 \\ \Phi(x,0) = x \quad x \in \mathbb{R}^N \end{cases}$$

How does one prove that $$\tilde \Phi(x,y,t) = \left(\Phi(x,t), \frac{\Phi(x + r y,t) - \Phi(x,t)}{r} \right)$$ is the flow of the ODE with $$\tilde{f}_r(x,y,t) = \left(f(x,t), \frac{f(x+r y,t) - f(x,t)}{r} \right)$$ as a vector field?

Also, in an answer to Prove that the flow of a divergence-free vector field is measure preserving, it was proved that if $\mu_t = (\Phi(\cdot,t))_{\sharp} \mu$ denote the image of the measure $\mu$ by the flow of $f$, then the family of measures $\{\mu_t\}_{t\in \mathbb R}$ satisfies Liouville equation $$ \begin{cases} \partial_t \mu_t + \operatorname{div\,} (f \mu_t) = 0 \\ \mu_0 = \mu \end{cases} $$ in the sense of distributions.

What PDE does $\tilde\mu_t = (\Phi_t)_{\sharp} \mu$ solve?

Suppose that $\Phi$ is the solution of $$\begin{cases} \frac{d}{dt}\Phi(x,t) = f(\Phi(x,t),t) \quad t >0 \\ \Phi(x,0) = x \quad x \in \mathbb{R}^N \end{cases}$$

How does one prove that $$\tilde \Phi(x,y,t) = \left(\Phi(x,t), \frac{\Phi(x + r y,t) - \Phi(x,t)}{r} \right)$$ is the flow of the ODE with $$\tilde{f}_r(x,y,t) = \left(f(x,t), \frac{f(x+r y,t) - f(x,t)}{r} \right)$$ as a vector field?

Also, in an answer to Prove that the flow of a divergence-free vector field is measure preserving, it was proved that if $\mu_t = (\Phi(\cdot,t))_{\sharp} \mu$ denote the image of the measure $\mu$ by the flow of $f$, then the family of measures $\{\mu_t\}_{t\in \mathbb R}$ satisfies Liouville equation $$ \begin{cases} \partial_t \mu_t + \operatorname{div\,} (f \mu_t) = 0 \\ \mu_0 = \mu \end{cases} $$ in the sense of distributions.

What PDE does $\tilde\mu_t = (\tilde\Phi_t)_{\sharp} \mu$ solve?