If you read my recent review of "X-Men: Days of Future Past," you know I had some issues with the movie's physics and metaphysics, especially with regard to Kitty Pryde's ability to project another person's consciousness back in time such that the present consciousness can inhabit that person's past-era body. As I noted, this creates the problem of what happens to the person's past-era consciousness: either it gets kicked out, in which case I have no idea where it goes; or it remains in the brain and gets temporarily(?) "overwritten" by the invading consciousness being pumped backward from the future, reasserting itself only after the future consciousness departs.
So what happens to Logan's mind(s) when Kitty Pryde sends him back to 1973 from the 2020s? And what about that question I had asked before: "What would happen if Kitty Pryde were to disappear? Would the future Logan's mind be forever stuck in his 1973 body?"
Second question first. I think I know the answer to this one: Kitty Pryde had to remain in constant contact with the 2020s-era Logan's body in order for Logan's mind to remain in the past. This implies that it was an effort to keep Logan's mind in the past; if Pryde's strength were to flag, Logan's mind would snap back to the 2020s. Why? Because in projecting Logan backward to 1973, Kitty Pryde was fighting the flow of time itself. Time naturally flows forward, so sending Logan to the Seventies was like struggling upstream in a river with a strong current. The current would never let up, so neither could Pryde. If Pryde ever did let up, Logan's mind would snap back into the future/present... but if Pryde has disappeared because the original future timeline has been erased, then where would Logan's mind snap back to? Would it automatically track down and inhabit alternate-Logan's mind, seeking Logan out like a guided missile? If alternate-Logan's current self already had a mind, having arisen from a completely different set of events, would alternate-Logan's consciousness again be kicked out (not in 1973 this time, mind you, but in the 2020s) in favor of original-Logan's? As we can see, the original question is answerable in the negative, given the strength of time's relentless flow, but other questions immediately arise as to where, exactly, Logan's mind would go.
As for the first question—what happens to Logan's mind(s) when he's sent backward?—there seem to be only two alternatives, already mentioned: (1) the past mind is ejected and goes God-knows-where, or (2) the past mind remains in Logan's 1973-era skull, but is "overwritten" by the 2020s mind, which is at best a temporary guest held in place by Kitty Pryde and her retro-temporal energy. Personally, I favor (2), as it seems to offer fewer loose ends. With (1), the problem is that we are again presented with multiple, messy alternatives:
(a) the 1973 mind simply goes nowhere—poof. Once ejected, it's gone, which means that once Logan's 2020s consciousness departs, there will be nothing left to inhabit Logan's 1973 body (at this point, I think we can assume that substance dualism is true in the Marvel universe; how else to separate a mind from a body?). A mindless Logan trapped in 1973 would further alter the past, thus further altering the future in very undesirable ways (assuming Logan's continued existence is a boon for the world).
Or (b) Logan's mind must go into some sort of metaphysical holding area, which takes us to the liminal space where science fuzzes into religion. In the Star Trek universe, certain types of consciousness can be physically contained, as happened on the planet Vulcan which, in JJ Abrams's 2009 "Star Trek," featured a very brief scene of a "katric ark" that allowed normal, enfleshed Vulcans access to the disembodied minds of the ancients and not-so-ancients.* Could such a hold exist in the Marvel universe, and would it serve as a temporary consciousness-container—a mind motel, if you will—for Logan's ejected consciousness? Or would the holding area be more like some "folded" region of space—less a brute physical container and more a subtle metaphysical one? Either way, the fact that we have to posit such a space in the first place indicates how unwieldy alternative (b) is.
All of this brings us back to what I'd said in my earlier movie review: it doesn't make sense. If you think too hard about the metaphysics of the X-Men universe, you'll just end up blowing a gasket. Better just to sit back, enjoy the movie, and not think.
*If I'm not mistaken, older Trek lore would style this the "Hall of Ancient Thought." Somewhat misleading, as recently deceased minds deemed great by Vulcan standards could also be ensconced there. In older Trek lore, each Vulcan soul, or katra, was placed inside its own katric ark, but the term as used by JJ Abrams refers to the entire structure carved into Mount Seleya, making it the equivalent of the Hall of Ancient Thought.