The question of what happens to a tathāgata after death is said to be avyākata or "unexplained". We know, negatively, that they do not undergo rebirth, since that is explicit, but we have no positive knowledge.
The question of how rebirth can work with no medium of continuity is not a trivial one. It has consumed Buddhist philosophers for centuries and we still debate the finer points.
For example, an orthodox Theravādin would say that each moment of mental activity is short lived and acts as a condition for the next moment. The last moment of mental activity (cuticitta) gives rise to a moment of consciousness somewhere else (paṭisandhicitta). But the Sarvāstivādins pointed out that the transition could not be instantaneous. The Theravādins had to insist it was instantaneous because any break in the stream would prevent the connection of causes to effects. The Sarvāstivādins had to invent a kind of interim realm (anatarābhava) which was not a rebirth, but which could account for the time it took for the citta to get from one to the next body). Unfortunately neither theory really explains rebirth. In the end there is no Buddhist description of conditionality that is entirely compatible with rebirth.
So you have hit upon a real weakness in Buddhist doctrine. And what we see is that in Buddhists texts where the message is about morality, a quite explicit personal continuity is implied that is quite at odds with the metaphysics of dependent arising. In fact Buddhists seem to switch back and forth between denying continuity and stressing the consequences of actions without noticing the howling contradiction.
The question of the five aggregates is also non-trivial. These have typically been seen as summing up all the physical and mental phenomena of existence (Nyanatiloka, Buddhist Dictionary). An important contribution to understanding the khandhas was made by Dr Sue Hamilton in two Books, the most accessible of which is Early Buddhism: A New Approach. In these books Dr Hamilton shared the results of studying every reference to khandhas in the Pāḷi suttas. Firstly they seemed to have nothing to do with existence. Dr Hamilton argued that the khandhas were more like the factors of experience. Indeed she influentially argued that Buddhist texts in general were commenting on experience rather than existence (or reality).
It can be useful to ask what the Buddhist texts refer to when they say that "things arise independence on causes". What is meant by things. A careful reading of Buddhist texts shows that was is mainly meant is dhammas - the objects of the mind sense (manas). Sometimes this is metaphorically linked to the world (loka), meaning "the world of experience" and sometimes to dukkha, which in this case is equivalent to unenlightened experience.
Now the Kaccānagotta Sutta (SN 12.15) says of this world of experience that talking about it in terms of existence (atthi) or non-existence (n'atthi). when experiences arise, it's not that experience exists or doesn't. It arises and it passes away. We have an experience and then it's gone. What was it that could have existed? Since the khandhas are just the factors, or even processes that make up experience, they have the same characteristics. The don't exist while you have an experience, or cease to exist when the experience stops. Experience is presence without substance. This is why experience is likened to a soap bubble, a dream, etc.
Nirvāṇa only really makes sense in an intellectual milieu in which rebirth is a given and seen as something to be avoided. But as prof. Richard Hayes once said:
“And if there is no rebirth, then the very goal of attaining nirvāṇa, understood as the cessation of rebirth, becomes almost perfectly meaningless. Or rather, nirvāṇa comes automatically to every living being that dies, regardless of how that being has lived.” (Hayes 1993: 13)*
For many of us who are engaged with both tradition and modernity this poses a major problem. It is apparent that there can be no rebirth under the known natural laws of universe. And yet this statement appears to disembowel traditional Buddhism and make it meaningless. And yet in the face of this theoretical argument, many people find Buddhist practices very helpful in many different kinds of ways. Whether or not there is rebirth, the practice of meditation, puja, mantra chanting, etc remains. As yet there is no very good way of explaining why we would still call this Buddhism. Maybe in the future we won't. Some people who think like this call themselves Secular Buddhists. Other labels are being tried.
- Hayes, Richard (1993). ‘Dharmakīrti on punarbhava’ In Egaku Maeda (ed), Studies in Original Buddhism and Mahayāna Buddhism. Kyōto: Nagata Bunshodo. Volume One, p. 111–30.