Here:
Like much of the news that ekes its way out of the totalitarian state, the murder is equal parts scary, sad, and vaguely comical.
I don’t think I had seen this extension of eke before.
Once upon a time, eke meant ‘also’; a relic of that sense is the word nickname, from an eke-name. (The transfer of the n from the article to the root was, I guess, favored by the alliteration.)
The phrase eke out a living meant ‘to supplement a fixed stipend’, as in The village priest eked out his meager living (i.e., the pay he got as priest) by making and selling strawberry jam. I guess that sense went away when the noun living itself got a broader sense; if your ‘living’ is your whole income, however obtained, you don’t add anything external to it.
So eke out (a living, or anything else) came to mean ‘obtain with difficulty’.
Information or water can be said to ‘find’ a way out of its container, but it seems a bit much to suggest that it does so with effort.