We fix some point $P_0$ and calibrate the speeds, so that $P_0$ doesn't move and stays still, say at point $O$, any other points moving. Choose some other point $P_1$ and let $t_0=0, t_1,t_2,\dots$ be the moments when $P_1$ meets $O$. Obviously $t_j=j\cdot d,j=0,1,2,\dots$ for some period $d>0$. For any other point $P\neq P_1$, there are two possibilities. Either $P$ meets only once $P_1$ at $O$ or it happens at least twice and it implies infinitely many times. If some point $P$ meets $P_1$ only once at $O$, then at that moment any other point should be also at $O$, in order the requirements to be fulfilled. So in that case, we are done. Thus, let any other point $P$ meets $P_1$ at $O$ at the moments $jd + i\cdot m d, i=0,1,\dots$ where $j,n\in \mathbb{N}$ depend on the point $P$. It easily follows there exists a moment all points are at the same place and moreover it occurs infinitely many times.

We can view the time $t$ being on the vertical axis projecting from the center $O$ of the circle. Then the motion of a point is represented by a spiral on the cylinder with base that circle. We know that any four of those spirals have a common point and want to prove all of them have non empty intersection. In fact, what the problem states is that the family of all spirals on the surface of a cylinder has Helly number $4$. I have written something about it here.