The philosophical question of personal identity over time is, roughly, the topic of what is required for a person at one time to be the same person at another time; due to some complications with questions of personhood, this is sometimes generalized a bit to the question of what is required for something (perhaps a person) at one time to be the same something at another time. (Olson's characterization in his SEP article, I think, gets this backwards, taking the latter as the fundamental problem; this is not tenable as an account of the actual topic of personal identity in the actual history of philosophy.) However, topics of this form are not exclusive to personal identity (even if we stay with speaking about persons). Here are just a few others of similar kind:
what is required for a person at one place to be the same person at another place
what is required for a person in one possibility to be the same person in another possibility
what is required for a person in one society to be the same person in another society|
what is required for a person in one doxastic state to be the same person in another doxastic state
what is required for a person in one legal system to be the same person in another legal system
what is required for a person with respect to one set of duties to be the same person with respect to another set of duties
More could be added. All of these, including times, can be modeled as possible worlds in one way or another, so all of these are in fact just different variations of what is known as transworld identity. Transworld identity is usually discussed in talking about some version of the second topic in the above list: what is required for a person in one possibility to be a person in another possibility. It has tended to be dominated by discussions of David Lewis's particular view of possible worlds, interpreted only as complete possibilities and literally as worlds, but as I've noted before neither of these interpretations are strongly motivated. You can perfectly well interpret 'possible worlds' as times or places or any number of other things, and occasionally we do.
Most attempts to give an account of personal identity over time focus on things particularly relevant to time -- persistence of a psychologically continuous stream over time, persistence of a material body over time -- but in reality, given that personal identity over time is really just a much more general problem applied specifically to the domain of times, any account of this should be something that is also generalizable, in one of two senses:
(1) the account for personal identity over times, or at least a generalized form of it, can be directly applied as an account in these other transworld identity topics;
(2) the account for personal identity over times is analogous to the accounts for these other transworld identity topics, in such a way that, while they have to be adapted to the different domains, the account for one can be used as a model for how to develop an account for another.
(In fact, the kinds of domains are diverse enough that I suspect that you will have (1) for some and (2) for others.)
When we try this out, most accounts of personal identity over time simply don't do very well. First, as noted before, they tend to be very time-focused, but it's not always clear what the analogue for the time-focused element would be in another domain. What is the analogue of psychological or physical continuity over time when we are talking about different possibilities or places or doxastic states? And what's more, even when you can find something that might be an analogue, it's often not clear that the analogue could even do the work it would need to do in the analogue domain to solve analogous problems.
Further, the answers often just push back any puzzles. If personal identity is a matter of a persisting material body, then we just have turned any questions of personal identity over time into questions of the identity of material bodies over time. This is not necessarily a bad thing in itself (presumably some of these transworld identity topics are in fact reducible to more basic transworld identity topics, in the sense that the interpretations of possible worlds may be more and less fundamental), but it's noticeable that the actual point in question -- identity over time -- is still on the board as a point in question. In fact, I think it becomes clear that most modern accounts of personal identity over time simply don't go deeply enough actually to give an adequate account of the topic. (In this way they are somewhat analogous to attempts to attempts to give an account of transworld identity that reduce it to some form of transtemporal identity. OK, but transtemporal identity is just transworld identity where the possible worlds are interpreted as times.)
