At least if I have no dependents, it would be praiseworthy, supererogatory and hence permissible to sacrifice my life to save the life of a stranger. It is permissible to take actions to save a stranger's life that I foresee (but do not intend) will cause my immediate death, and I equally have the right to take actions to save a stranger's life that will result in my dying in 20 years (e.g., because to save the stranger's life I need to brave some radiation hazard which will eventually kill me.
This is true even if I foresee that my character will shift in significant ways over the next 20 years, that I will then not have many memories of how things are for me now, and so on. In other words, I have the right to sacrifice the life of my future self for a stranger even if my future self will not be very much psychologically connected to me. On the other hand, I have no right to sacrifice the life of my friend for a stranger without my friend's permission, and the closer the friend is to me, the worse such a sacrifice would be. This shows that my relationship to my future selves is significantly unlike my relationship to my friends. It does not matter how psychologically close to or distant from my future self I am--I have the right to make the sacrifice. But the closer I am to my friend, the worse it is to sacrifice the friend for a stranger without the friend's permission.
Thus, sometimes, identity matters, and psychological connectedness is largely irrelevant. Could one replace identity with psychological continuity in these considerations? No. For suppose that I know that next week I will fission into two individuals. I will have psychological connectedness and continuity with them. But I have no right to sacrifice their lives to save the lives of two strangers, e.g., by exposing myself now to a dose of radiation that will result in the two descendant individuals dying in two weeks. It would be like sacrificing the lives of two children of one's own to save two strangers. The psychological connectedness and continuity are insufficient for permissibility here. And even if it were permissible, it would hardly be praiseworthy.
Or consider the following scenario. Rescuing the stranger will result in a dose of a substance to oneself that will first induce amnesia and then death. On psychological continuity theories, the person who dies is the person who inhabits your body after the amnesia and this is not you because of lack of psychological continuity (one might add that one has no plans for this person if need be). And so you have sacrificed the lives of two people to save the stranger--for you will die through the amnesia (on psychological continuity theories, amnesia is a cessation of existence), and then the new person who will come to exist in your body will die. So this is imprudent (sacrificing two to save one) and immoral (one of the sacrificed has not been consulted). But that is absurd--surely it makes no difference whether the substance directly causes death, or causes amnesia followed by death.
(In fact, the latter consideration seems a nice independent argument against psychological theories. Suppose that you're dying. Treatment A will give you 100 days of life. Treatment B is slightly less painful, and will result in 100 days of life, followed by amnesia, followed by an hour of life, followed by death. It seems permissible to opt for Treatment B. But if Treatment B means two people die--one through amnesia and another after--it may not be permissible to opt for it just because it is slightly less painful.)