Conditions on change of variables for integral constructed as flow of ordinary differential equation

Question

Consider an integral \begin{equation} \mathcal{I}=\int dt \int d^3 x f(x) h(t) \end{equation} and the change of variables $x=\phi(t,x')$, where \begin{align} \partial_t \phi(t,x')=F(\phi)\\ \phi(0,x')=x'. \end{align} In principle the change of variables is well defined on all points which are not critical for the vector field $F$. However, I'm pretty sure that if we adopt Lebesgue's definition of integrals and exclude all the critical points of $F$ from the integration (which should lie on zero measured surfaces, if $F$ is continuous), then under some regularity conditions on $F$, the integral is still well defined. More concretely, what I'm looking at, after the change of variables, is \begin{equation} \mathcal{I}=\int dt \int d^3x' |\text{det}[\mathbb{J}_{x'}\phi]|f(\phi(t,x'))h(t). \end{equation} and I'm wondering what conditions (and why) need to be imposed on $F$ so that the change of variables is well defined and integration is convergent. Assume $f$ and $h$ are $L_2$, for simplicity.