Thinking why would Claire stay on the 2004 timeline when the time shifts occured (as she was in treat of The Others), I came upon this theory:
Only people who are candidates are able to travel through time.
This would explain why Claire didn't, as she turned to MiB's side, joining him in Jacob's cabin, and would explain why did Frank, Ben, Sun, Ilana and Richard stood on 2007.
- Frank is not a candidate, as far as we know.
- Ben could perfectly be crossed out already, as he was not loyal tons of times (or it could be Roger)
- Sun.. she may not be a candidate, and instead, it is Jin, who travelled through time (and met Rosseau).
- Ilana also stood at 2007 because she is not a candidate (see her recently discovered surname on episode 13).
- Richard's status is the same as Ilana's.
- The Others may not include any candidate at all, so it is applied to them too.
The other losties that haven't made out of the island and experienced time flashes would perfectly all be candidates at that time, where Jacob was still alive:
- Sawyer is a candidate, with the number 15
- Charlotte, Daniel and John Locke all have their names crossed out, but at that time, they probably were still candidates.
- Miles and Juliet.. I can't explain why their name were crossed out and still travelling through time... (note that Juliet died after Jacob)
- Jin is still a candidate, and I explained his difference between Sun up there
Now, about the ones who were on the plane and did travel:
- Hurley is a candidate, number 8
- Sayid is a candidate, number 16
- Jack is a candidate, number 23
- Kate is a candidate.. on the lighthouse she is not crossed out (which I see as Jacob's) while in the cliffside cave she is crossed out (which I see as MiB's). MiB also said to Claire that Kate is not a candidate and he does need her to get all the candidates and only that, which explains why her name is crossed in a certain way. (Both lists of candidates have differences)
- Locke's body, as a deceased body, stays in the main timeline, like Charlotte did when she died.
My only problem is Miles' and Juliet's status... there has to be something between the end of time flashes and the incident that caused them to be crossed out EDIT: Bernard, Rose, Vincent and the background survivors also travelled through time, so my theory does not apply to them!
- Although Rose was a possible candidate, she is crossed out with no clear reason
- Vincent probably has no surname, and I really doubt he is suposed to be a candidate at all