I have looked up literature on omega limit sets and from what I have seen they consider limit sets of a single point (of a single trajectory, discrete or continuous).
Are there studies that characterize structure and properties of total $\omega$-limit sets, meaning the union of $\omega$-limit sets of all trajectories?
The union of all the $\omega$-limit sets of a function $f:X \rightarrow X$ $\,(X$ is a metric space) is sometimes denoted by $\Lambda(f),\,$ that is, $\,\Lambda(f) = \bigcup_{x \in X}\,\omega(x,f).$ See, for example, The persistence of $\omega$-limit sets defined on compact spaces by Emma DʼAniello and Timothy H. Steele (2014). I recommend sending an email to Steele (feel free to say I told you this, as I know him), asking him for some pointers to the kind of results you are interested in pursuing.