Basing this more on fiction than anything (but at this point, what's the difference) I prefer the "always was" type of time travel - Kyle Reese goes back in time to protect John Connor('s mother) but becomes his dad in the process. Who was the original father? Doesn't matter, that has been changed and not it always was that Reese was his father.
T1 indeed heavily implies that meta-time is static, in its universe, that it was a self-serving timeloop with no evitability (though Reese is apparently not made aware of this; possibly him being fully aware would not support the closed curve the cosmos needs, so from among all candidate cosmii the 'solvable' one we see here is one where he is not told, even though clearly John found out ahead of (future) time). See also Twelve Monkeys. And, interestingly, Bill & Ted.
T2/T3 seems to suggest mutability of the future. Though if you assume 'dishonest' (or, rather, deliberately misinformed) future knowledge brought back to the present day, the 'avoided' future could be a fiction the true future individuals (human and otherwise) knew were part of the mix of information that
had to be inserted into their past. This implies the temperal-theoriticians involved knew this enough to not even
try to remake their history. Even Skynet, who sent back various 'failed' missions in order to support the situation in which Skynet does
not have its own perfect machine-utopia. But then maybe Skynet is just the most logical (and willing to maintain the meta-time loop) of all parties, and is actually ultimate caretaker of ensuring the cosmos ticks over the way it clearly does. There
are ways of the whole movie franchise to this point to be fully self-consistent, even when some (within it) believe otherwise.
From what I recall, Sarah Connor Chronicles threw that completely out (but I didn't see the whole series[1]) by making it a pliable-timeline universe in demonstrable ways. Which conceivably (NPI) might work just as well with John's movie-origin story (i.e. less self-bootstrapping than
All You Zombies, but clearly
informatically near-indistinguishable lineage each time round) but, as with othe Film-to-TV adaptations (e.g. Stargate) can probably be considered a different cosmos entirely with different rules in some aspects of reality.
[1] It was awkward to watch, as I recall, both when originally hoving into view and in later repeats, I kept seeing just the same half-dozen episodes over and over, and never did get around to look for tapes or torrents or whatever of the whole thing.
I have yet to see any fiction deal with [space-synchronisation across temporal movement] adequately. I suppose if we have that sort of travel capability, we would more likely explore the stars rather than go back and get revenge on history's greatest monsters.
The usual answer is just being tied to the same spacial Frame Of Reference as you propel (whichever way) through the temporal one. H.G. Wells's eponymous device sat in the same physical spot while travelling[2], so tied to the given shifting techtonic plate, upon the spinning Earth, in its orbit around the Sun, in its path around the Milky Way, in its route within the Local Cluster, as that itself travels as required by the forces in the respective supercluster(s) and also however the expansion of the universe itseld interacts with any atemporal drifting. If the Machine were not sat in a workshop but in an orbital position, or some other 'free' trajectory (even whilst still going at 1s/s existence without any special movement ±
t-wise), it probably would not be tied so closely to the continuous view (for as long as possible) of the changing fashions in the shop window opposite.
Or, for those not yet ready to accept any Wellsian space-travel in that setting (or at least
a derivative one), imagine you mounted your time-steed as it sat in a train goods van (or atop a flatbed, for that extra whistling wind) and then manipulating the controls to take you along in either direction of forced time, but are you 'tied' to the carriage? Go ahead three years, and you are where the (latest towing) train has moved your 'ground' to, back three years and you'd find where it had been. Before and after its true existence as a bogied and bodied platform do you find yourself within the initial fabrication location (or where it would later be built) or the scrapyard (or what comes to be once the scrap is also removed)? It might assist tue understanding of fundemental Natural Philosophy to
try such a machine upon the decking of a Ship Of Theseus, assumingbone were provided with full working warranty but absent any explanation (or sufficiently annotated manual) from the device's designer/constructor.
If not entirely a 'thing that the Universe does for you', constructed time-devices may have specific spacial-lock (or synchronisation) elements in their mysterious internals (such that, where plot has allowed but now requires it not to be, the
secondary level of spacial control - that adjusts where the "autohover" facility thinks it should 'stick' to - can be made inopetable such that the user has no power to control anything
other than the time movement, however much the unit has to strain to maintain 'formation' with the evolving terrain (perhaps accounting for eroding bedrock or accumulating silt, so as to at least 'ride' the geology).
Or, any attempt at (propulsive) time-travel just sets 'here' as some frame completelg dissaciated with the substrate (perhaps the 'absolute inertia', instantaneously upon departure, is extrapolated forward/backward through space with no contact or gravitational forces applying in your transitting stats, thus flinging yoy off-planet and off to where you'd end up (or have come from) in interstellar/intergalactic space if you had instead just designed an "all external matter vanishes" pod and waited/been waiting for the requisite time of freefall.
And I'd definitely say that this issue
has been addressed in fiction. Maybe in a Handwavium/technobabble way tuned to suit the fiction's own conceipt (to support its intended plot), but while many works stick with the implicit "it just works", others at least
pretend to construct a viable operational theory.
For non-vehicular time-transportion, like wormholes[3], the mechanism is whatever it is that 'anchors' (and/or tows) the respective wormhole-ends. Either as part of its natural evolution or in deliberate manhandling/installation by the creators/exploiters of the phenomenon.
[2] Ignoring "outside world is visible as times are passed through, but the 'travelling' machine isn't visible as passing through(/existing in, perhaps as a frozen 'shade') those times where there should be witnesses" - also in going forward and backward, in the same location, are you also co-existing with your other self(/ves) as well as the location travell8ng along 'natural' time. But Herbert George was very much new to this (then) very much new game, so we can forgive the odd question like this.
[3] Not counting the vehicle you might need to use to traverse the hole, if it has no time-propulsion of its own, and is just being used to safely slide down/up the time-chute.