I am trying to understand why the (positive or omega) limit set of a non-autonomous dynamical system \begin{equation} \dot{x}=f(t,x) \label{ftx} \tag{1} \end{equation} is not necessarily (positively) invariant. Assume that $f$ is piecewise continuous in $t$ for each fixed $x$, and that $f$ is locally Lipschitz in $x$ uniformly over $t$.
I have seen the proof for the autonomous case in this question, and I imagine that, for the non-autonomous case, a set $\bar{\omega}$ may belong to the positive limit set of \ref{ftx} for a given initial time $\bar{t}$, but if the initial time is changed to some $\hat{t}$, then it is possible that the solution will enter $\bar{\omega}$ and leave it.
Am I correct? Or is there something I'm missing that will break in the aforementioned proof?