This is not an answer, but rather a long comment. Software like Mathematica can plot trajectories of this dynamical system, by (numerically) solving the system of ODEs. Here are trajectories that start on a circle of radius $1/10$ centered around the lowest nontrivial zero of $\zeta(s)$ (sorry, I can't make myself call the variable $Z$.) [![enter image description here][1]][1]
The picture near the zero is exactly what you expect when you think of approximating $\zeta(s)$ by the linear approximation: the ODE in $s=\sigma+it$ is very nearly $\sigma^\prime=\sigma$ and $t^\prime=t$, at least, up to the scaling and rotation of the picture by $\zeta^\prime$ evaluated at the zero. Further from the zero, higher order terms come in to play.
Here are trajectories that start on a (larger) circle of radius $1/2$ centered on the pole at $s=1$. This picture is also what you should expects when you think about the Laurent expansion of $\zeta(s)$ at $s=1$
[![enter image description here][2]][2]
I feel like I should know the answer to the OP's question, not just for $\zeta(s)$ but for meromorphic functions in general. Don't the Cauchy-Riemann equations tell us something?
I don't believe there are any closed orbits. [1]: https://i.sstatic.net/xWsxN.jpg [2]: https://i.sstatic.net/6LQMI.jpg
