I think this is a good question, generally neglected by religious practitioners. If you go to heaven, what goes to heaven? If you are reborn, what do you have in common or what continuity is there?
Even most Christian's don't know their own doctrines, about bodily resurrection, and a new Heaven and a new Earth. The idea of immediate transmigration into Heaven, is for saints. It occurred in a cultural context of Hellenic ideas of ascent to a position of being remembered and called on for guidance, through having constellations named after you (like Hercules and Perseus did). Discussed here: What are some philosophical works that explore constructing meaning in life from an agnostic or atheist view? The Jewish perspective on Heaven is quite different, in that Shabbat is literally practice for it - it's much more like making peace with your life, finding a way to be that you could be happy with for eternity.
In Buddhist thought the continuity of identity, is much less than practitioners generally reflect. The core doctrine of Anatta specifically rejects the idea of an inner essence providing a clear link between beings. The rebirth of causes and conditions, is more like using one candle to light another and how much the candles then have in common - as per the metaphor in The Questions of King Melinda, discussed here: Does Buddhism espouse reincarnation? Buddhist thought deconstructs the conventional self, as a tool to witness Sunyata, which Thic Nhat Hanh called 'Interbeing', the interdependence and conditionally of things. If you had experienced the causes and conditions of someone else, you would be them. So there is a deep commonality between us all. Discussed here: How would you apply John Rawls "Theory of justice" to everyday decisions?
For me the really important insight is about intersubjectivity. This is made really clear in the ancient Indian metaphor of Indra's Net. But it's also critical to understanding how morality goes beyond the game-theory of getting what we want. Discussed here: Is the Categorical Imperative Simply Bad Math? :)
The most detailed picture of rebirth I think, is the Yogacara Mahayana doctrine of Alayavijnana, or Eight Consciousnesses. This is an elaboration of the sense-gates picture of the arising of consciousness from the modes of interaction of each sense, and the associated domain or space of the information provided by it. Then the sixth 'mental consciousness' is very like Aristotle's picture of a Common Sense, where different senses construct integrated objects in the mind. The seventh is translated as 'deluded consciousness', we can relate to opinions, which Buddhist thought holds as problematic ("If you wish to see the truth, then hold no opinions for, or against, anything. To set up what you like against what you dislike is the disease of the mind."-3rd Zen Patriarch in the HsinHsinMing. Eighth is translated as 'storehouse consciousness, and I think this bears very good comparison to the Noosphere, or Memesphere. That is, the space of ideas which have their own ability to persist and achieve substrate independence. Taking this seriously, we can come to an understanding of rebirth that can be much better reconciled with science.
The picture I am increasingly drawn to, is more like the idea of Eternal Recurrance, in Nietzsche's thought. In a Multiverse, we can get an expanded picture of us following all our possible paths in life, and of all the very similar selves with subtly different initial conditions. In such, we can picture the movement towards being reconciled with our lives, to not wishing to go back and change things, to recover ideas about Bodhisattvas and their buddhasetra, and of Grace. Non-attachment as finding a way to live, being reconciled with our choices moment by moment, so we can at last let go with tinkering with do-overs of our life, or the lives of others only similar to us from wishing for changed initial conditions (see Teletransportation Paradoxes and the divergence of selves). With the static-block picture of spacetime, we could imagine rebirth as the momentary standing outside of the subjective experience of rolling through the moments of our choices looking at the structure as a whole and wishing for change. So the aim of rebirth as something like, creating a 4D sculpture you are happy with, to build a Good Life. Then we can truly let go, and go:
"Beyond, go beyond, go completely beyond. So be it." -closing mantra
of the Heart Sutra