My understanding of the Theravada perspective:
what is passed from life to life in the process of rebirth?
Depending on what one means by "passed", the stock answer is there is nothing passed from one life to another.
If something passes from one life to another (be it memories, consciousness, karma, etc), than that something should be regarded as one's Self. However, the Buddha's admonished (sometimes, strongly, see MN 38) when someone say his teachings promulgate that some substance can be observed to go through rebirths.
However, the Buddha also said we are heirs of our karma:
“ Bhikkhus, beings are the owners of their kamma, the heirs of their kamma; they have kamma as their origin, kamma as their relative, kamma as their resort; whatever kamma they do, good or bad, they are its heirs.
-- AN 10.216
The difficulty with grasping this subject intellectually is the doctrine of anatta/no-self and buddhist doctrine of karma. Thus, the puzzle:
- if there's no substance that persists, how could we say that something passes from life to life?
- But, if there's no such substance, why the Buddha rejected the idea that physical death is simply the end and talked about "beings appearing at X after death"? And what does it mean to "inherit" karma?
Generally speaking, if we say that karma is what is passed from life to life, then we are turning karma into a substance that persists: saying this is roughly the same as making karma the Self, the Soul. Thus, many avoid saying something of the sorts.
Consider Bhikkhu Bodhi's words about the word "rebirth":
The Buddhist term for rebirth in Pali is "punabbhava" which means "again existence". Buddhism sees rebirth not as the transmigration of a conscious entity but as the repeated occurrence of the process of existence. There is a continuity, a transmission of influence, a causal connection between one life and another. But there is no soul, no permanent entity which transmigrates from one life to another.
So, an answer here is that there's no substance that is passed from life to life, instead there is a causal connection, a process of influence. If the fetters have not being put to an end, after death there's continuity of experiencing dukkha, continuity of experiencing samsara.