Let $f$ be a $\mathcal{C}^1$ vector field (VF) on a compact subset $M \subset \mathbb{R}^n$. $M$ inherits the Euclidean metric. We define a dynamical system by $$\dot{x}(t)=f(x(t))$$ The dynamical systems is topologically equivalent to another systems (with VF $g$) if we can map orbits to orbits, preserving the time direction. More precisely on the level of VF there has to exist a diffeomorphism $h: M \rightarrow M$ such that $$f=(Dh)^{-1}(x)g(h(x))$$ with the Jacobian $Dh$ of $h$.
Given some distribution of initial conditions $\rho_{0}$, the time evolution under the flow of $f$ is defined by
$$\left\{\begin{array}{ll}\rho_{t}+\operatorname{div}(f \rho)=0 & x \in M, t>0 \\ \rho(x,0)=\rho_0 & x \in M\end{array}\right.$$
Let $\mu(t,x)$ now be the evolution of $\rho_0$, (i.e. $\mu(x,0)=\rho_0$) but under $g$ instead of $f$.
My question is the following: Can we establish a relation of $\mu$ in terms of $\rho$. I.e. can we write $\mu$ in terms of some (closed) expression involving $\rho$ and $h$.
I am also happy to anybody pointing me to the literature, if a related problem was already studied.