This is an edited version of a HUM331 forum post which I made I to follow up on a class discussion about how rebirth is possible given non-self (originally with the particular context of Buddhist hells). The quotations here are the most relevant views I could find in the books I had on hand. See my other post for my own answer to the related question of whether karma and rebirth should be considered mythological or metaphorical as opposed to literal.

Walpola Rahula quoting Shakyamuni Buddha

In What the Buddha taught, Walpola Rahula says (chapter 6, page 66)

Here naturally a question arises: If there is no Atman or Self, who gets the results of karma (actions)? No one can answer this question better than the Buddha himself. When this question was raised by a bhikkhu the Buddha said: "I have taught you, O bhikkhus, to see conditionally everywhere in all things." [footnoted as Majjhima-nikaya III (PTS edition) p19, Samyutta-nikaya III (PTS edition) p103]

I'm not very familiar with the Pali texts and wasn't able to track down the context.

Buddhadasa

Buddhadasa has what I find to be the clearest explanation. In No Religion, he says (pages 11-12)

In people [ordinary] language, the word "birth" means to be born from a mother's womb. In Dhamma language, however, the word "birth" means some form of attachment is born. This kind of birth happens every time we allow the arising of a thought or feeling which involves grasping and clinging to something as "I" or "mine," such as, "I am," "I have," "I think," and "I do." This is the birth of the "I" or the ego.

For example, think like a criminal and one is instantly born as a criminal. A few moments later those thoughts disappear, one thinks like a normal human being again and is born as a human being once more. If a few moments later one has foolish thoughts, right then one is born as a fool. If one then thinks in an increasingly foolish and dull manner, one will be born as an animal immediately. Whenever an attachment is felt intensely--when it burns inside one with the heat of fire--one is born as a demon in hell. Whenever one is so hungry and thirsty that one could never be satiated, one is born as an insatiably hungry ghost. When one is overly cautious and timid without reason, one is born a cowardly titan. Thus, in a single day one can be born any number of times in many different forms, since a birth takes place each and every time there arises any form of attachment to the idea of being something. Each conception of "I am," "I was," or "I will" is simultaneously a birth. This is the meaning of "birth" in Dhamma language. Therefore, whenever one encounters the word "birth," one must be very careful to understand its meaning in each particular context.

[...] whenever an egoistic thought or feeling arises, there is suffering at once and the suffering always fits the particular kind of "I" that is being born. If "I" is human, it suffers like a human. If "I" is an angel, it suffers angelically. If "I" is demonic, it suffers hellishly. The manner of the grasping and clinging can change repeatedly, even being born as beasts, hungry ghosts, and cowardly titans. In one day, there may be many births, many dozens of births, and every one of them is unsatisfactory, frustrating, and painful. To destroy this kind of birth is Nibbana.

Concerning death, there's no need to speak about what happens after the people language version. Why talk about what happens once we're in the coffin? Instead, please deal with this most urgent issue of ego-birth, that is, don't get born and there will be no suffering.

In Heart-Wood from the Bo Tree he further explains (page 26)

In one day there can be hundreds of births; the amount depends on a person's capacity, but in each birth the "I" and "mine" arises, slowly fades, and gradually disappears and dies. Shortly, on contact with a sense-object, another arises. Each birth generates a reaction that carries over to the next. This is what is called the kamma of a previous life ripening in the present birth. [...] For example, there can be birth as the desirer of some pleasing object and then death followed by birth as a thief or robber, and then a further death followed by birth as the enjoyer of that object. [...] If there is still birth, still the feeling of "I" and "mine", it just goes on being the Wheel of Birth and Death, a continual chain of Dukkha.

At the retreat I did at Buddhadasa's retreat centre we were taught these same views mostly in the context of the twelve-point cycle of conditioned arising. The Wheel of Birth and Death referred to above incorporates this cycle. The cycle can be viewed either as describing the process of birth and death in the conventional sense, or as describing moment-to-moment rebirth. The later is what Buddhadasa focuses on above, but his underlying message as I understand it is that since there is no self apart from self-thoughts, these two views are not in fact separate.

Sheng Yen

Sheng Yen indicates that karma is what persists between lives. In Subtle Wisdom he says (chapter 3, page 41)

Buddhism does not believe in any permanent, unchanging soul or spirit, but holds that some karmic seeds planted by our actions in this life will be reborn in order to bear fruit in another life.

The surrounding context clarifies

As we move forward in time, we incessantly create future causes and future effects. We create our future suffering by what we do now. "Karma" means action, and it refers to both cause and consequence as they produce each other in this cycle. [...] The causes and effects, driven by desire, are, themselves, states of suffering.

Note that he describes karma simply as action here. Sheng Yen often talks about causes and conditions instead of talking about karma, but they seem to be equivalent in this case.

In There Is No Suffering he says that the fifth skandha "is what generates future lives" (chapter on "The Twelve Links of Conditioned Arising", pages 75-77). This is given in the context of discussion of the cycle of conditioned arising. In Orthodox Chinese Buddhism he says that the eighth (storehouse) consciousness "is the entity which integrates the life process" (2.2, page 35-37) between lives. Interestingly this description initially starts out with an answer very similar to the one that Shakyamuni Buddha gives in the quote above, beginning "Buddhists believe that 'phenomena arise dependent on conditions' [...]". In Setting in Motion the Dharma Wheel Sheng Yen says (chapter 2)

According to the agamas and the Abhidharmakosha, there is another dimension of meaning to the five skandhas, namely, 'grasping.' Grasping arises when a sense faculty interacts with a sense object, creating attachment, and consequently, suffering. This grasping after sense experience assures the continuation of the five skandhas through life after life. The objects of grasping are not just desires, but also hatred and delusion. Simply put, grasping causes suffering and in turn, suffering causes the continuation of the five aggregates through rebirths. On this basis we hold onto the poisons of greed, hatred, and ignorance which propel us into future rebirths. Then, because of the five skandhas, we give rise to vexations again. So vexations cause the five skandhas, and the skandhas cause vexations. They are inseparable, mutually causing each other.

In the video "Conditioned arising induced by karma" on the DDMTV05 Youtube channel, he indicates that various views of conditioned arising are just different ways of referring to the same thing; I suspect that applies to the present topic as well.

I think it is important to note that in the views of the fifth skanda and the eighth consciousness as persisting between lives, these are specific parts of consciousness (in particular analyses) and are themselves subject to impermanence and non-self. For example, in the section on the eighth consciousness Sheng Yen notes that this consciousness "exists in the continuum of momentarily changing karmic seeds and fruits" so that it is "not equivalent to an eternal soul" (page 37). Moreover, he describes both the fifth skanda and the eighth consciousness in terms of karma.

Note that the mentioned part of There Is No Suffering is in a section on the cycle of conditioned arising, and the paragraph from Setting in Motion the Dharma Wheel uses similar language (compare to Buddhadasa's description). Although Sheng Yen does not take the moment-to-moment view of dependent origination in describing rebirth, he does indicate that seeing through the cycle can end suffering immediately: "the realization of the un-arising nature of the twelve links is cessation of suffering" (in There Is No Suffering, same chapter, page 78).

Non-birth and non-death

One Buddhist view is that there is fundamentally no birth or death. For example, the Heart Sutra that says "all dharmas [...] are neither born nor destroyed" (Sheng Yen's translation in There Is No Suffering). Commenting on this verse in Lok To's translation of the Heart Sutra, T'an Hsu says

In the Madhyamika Sastra, Bodhisattva Nagarjuna says: "For the one who is already born, there is no birth; nor is there birth for the one who has not been born. The one who was born and the one who was not born, neither has birth-nor the one being born has birth at the time of his/her birth." To give an example, grass that is one foot tall is no longer sprouting. That is what is meant by "no more birth for the one already born." Supposing the grass that is presently one foot tall is allowed to grow one more foot: It still cannot be said to have birth, because there is no manifestation of birth. That is meant by "what has not been born yet has no birth." The grass cannot be said to "have birth" or "being born" at any specific time during its sprouting and so it is said that "the one being born does not have birth at the time of birth."

[...] For the one already dead there is no death; for the one not dead yet there is no death, either. At the time of dying there is not one specific point in time for death to manifest itself.

Sheng Yen, commenting on the same verse in There Is No Suffering, says "every moment, your life -- indeed, the entire universe -- is a dynamic process of birth, growth, decline, and death" (chapter on "Impermanence", page 55).

This view seems consistent with and perhaps implied by both Buddhadasa's focus on moment-to-moment rebirth through self-thoughts and with Sheng Yen's descriptions of karma or karma-based consciousnesses persisting.