A ship travels 99% of c from Earth (leaving at Time A, say 2016) to Destination Star (arriving at Time B, say 2020, accounting for relativistic effects) to stretch the tube, leaves a buoy at Destination Star, then uses the tube's topology to return to Earth, effectively faster-than-light, moments after it left, according to how the Krasnikov tube is supposed to work (Time A plus a few minutes).
It just can't work like this. Think about what a clock actually does. Time is a measure of local motion, that's all. It's "how much local motion has occurred". If you move very fast, the rate of local motion is of necessity reduced because the maximum rate of motion is c, because of the wave nature of matter. Hence you suffer time dilation. See the simple inference of time dilation on Wikipedia. Note that if you have two parallel-mirror light-clocks and one stays at home whilst the other goes on a fast out-and-back trip, the light-path lengths are the same. And they meet back up at the same time. They don't miss each other by a week. Time dilation is not time travel.
Can the ship now return to Time B (2020) from "Time A (2016) plus a few minutes" using the tube just as quickly?
No, it absolutely can not. At .99c the Lorentz factor is 7, so if your journey takes you four years by your clock, your destination star is 28 light years away. When you get there it's circa 2044 according to Earth clocks. There is no way you can get back to Earth in 2016. For that to happen all the motion that occurred has to be undone. And there is no way that that can happen.
If so, would its return to Time B mean "Destination Star in the ship's year 2020, the moment when it arrived / first stretched the tube to this destination?" And if so, would it mean returning to the system "before" the ship dropped the buoy? My understanding of a Krasnikov tube is that it's a theoretical means for FTL travel that doesn't violate causality, because a ship traveling along the return path of a tube "rolls back" time to just moments before the ship originally left.
It's just science fiction I'm afraid, with no basis in physics.
Now, I understand that there is a way to construct a return tube that would violate causality (http://arxiv.org/pdf/gr-qc/9702049.pdf), but let's assume for a moment there's some "chronology protection" in place that doesn't allow that to happen (the tube implodes when you try to do that), and that the question of the exotic matter needed to create such a tube isn't an issue for the ship's makers.
The chronology projection conjecture is yet more speculation I'm afraid. It's superfluous because we don't actually move forward through time. Hence moving backwards through time is a non-starter.
Am I even thinking about this correctly? Where have I messed up?
You didn't watch carefully, and you fell for the smoke-and-mirrors and the sleight of hand.
I don't understand how a system of Krasnikov tubes, as some sources suggest, could be used to create an interstellar network for FTL travel. It seems to me that the traveller in the ship can never actually have an impact on his exit or return points, since he will always be returning to Time A (2016) moments after he left, or Time B (2020), the moment he arrives?
If he can go from 2044 to 2016, it's a time machine. Only he can't, so it isn't.
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