The derivation of Liouville's theorem (for example this one) makes use of the concept of "movement" of the points in the phase space.
Let it be a system of $N$ particles, with $q=\{q_i\}_{i=1}^{3N}$ the generalized coordinates and $p=\{p_i\}_{i=1}^{3N}$ the generalized momenta of the $N$ particles. If $(q,p)$ is a point of the phase space (also called representative point), a velocity vector $\vec{v} = (\dot{q}, \dot{p})$ is defined for $(q,p)$. This velocity, as time passes by, will give the direction of the trajectory of that point in the phase space.
This idea of the motion of representative points is something that confuses me, since in the three-dimensional space, for instance, the points $(x,y,z)$ of the space do not move anywhere (the point $(-3,9,1)$ will remain located at $(-3,9,1)$ in saecula saeculorum). In any case, what will move is the position vector $\vec{r}(t)=(x(t),y(t),z(t)$.
How should we understand the idea that, with time, representative points moves, entering and leaving a $\omega$ volume of phase space?