Looking at a Wikipedia article describing a telefilm, I noted the following sentence that describes an event happened in Season 1:
They eventually find and reunite with their father, who reveals that the creature that killed their mother years earlier is the demon Azazel (aka "Yellow Eyes") and the only thing that can kill him is a legendary gun created by Samuel Colt.
I am used to sentences similar to the following one. (Suppose somebody asks me what my work plans are.)
I want to get a job on the local newspaper and eventually work for "The Times."
I understand eventually as introducing an eventuality: It can happen that I will work for The Times, but it could also happen that I will not work for The Times. The previous sentence is different from "I want to work for The Times."
Probably I am influenced from how in Italian we use eventualmente, but I don't understand what eventually would mean in a sentence describing what happens in a telefilm season.
Best Answer
It means after a long time, or after a lot of effort. See eventually in this online dictionary.
So in your first example I imagine they have to go through an extended period away from their father, or they have to do a lot of work to find him.
(in response to a comment) It is possible to say "Eventually, I will come back tomorrow morning." but only in a very convoluted scenario, where tomorrow morning is a long time away, e.g. in a time travel scenario.
Eventually carries the feeling that there will be a struggle or a need to be patient.