I would argue that asking such a question is not appropriate for our site. Collecting answers to overly broad questions is not what Stack Exchange is meant to do, nor what it does well.
Just because a question is marked Community Wiki doesn't mean it doesn't need to adhere to our quality standards. As bilbo_pingouin already pointed out in the comments, a question such as the one you propose would be extremely broad. (You are asking for all possible mechanics and without bound their narrative consequences, for all possible imagined faster-than-light propulsion systems. This one wouldn't even be limited by, say, a science-based or even a reality-check tag. If that's not closeworthy as "too broad", then frankly I don't know what would be.)
I could design a FTL drive for my universe which is powered by the digestive byproducts of little white mice eating bits of cheese after running through labyrints, and describe that system including its narrative consequences in excrutiating detail. (The probability of such a system? Most likely, it would be almost infinitely improbable. But who are you to judge?) It isn't likely to be useful to anyone else, but it would be a valid answer to your question as proposed.
In addition to the above (which is largely generic), as far as the specific example is concerned, almost any FTL-type drive would have similar consequences and narrative implications. This includes allowing violation of causality. The specific technological gizmo or handwavium used to overcome this might well be different from one story to another, but the problem to be overcome (or simply handwaved away; or not overcome, if that's what you want!) remains exactly the same.
You can, however, ask a series of questions that are sufficiently specific for our site, and then post a reference meta question about the series. I'm not entirely thrilled about that latter either myself, but that is much more a valid use of the site. If you go this route, then please spread the questions out over time so that each can get the attention it deserves.