At the time of death the body dissolves.
After the dissolution of body, self without body remains.
Why this self remains?
Because of volitional activities.
Whenever a volitional activity is made, self appears.
It is at that time that this self, by volitional activities, sustains and destines himself to a future rebirth.
"If we the take the simile of lamp then my question would be: who
passes the flame from one lamp to another ?"
To a liberated mind, the answer is nobody.
To a liberated mind, there is nothing to pass.
There is no flame, no lamp.
They are mere fabrications ... not truths ... leading to fabricated things ... lies, that seem like truths. Destined to cessation. Leading to impermanence. Leading to craving and clinging on these impermanent lies that seem like truths, thus seeding seeds for future suffering when they cease.
Knowing this truth, a liberated mind does not fabricate fabricated things ... his volitional activities cease ... his kamma exhaust ... self does not appear, thus no rebirth happens.
To a not yet liberated mind, the answer is "self".
To a not yet liberated mind, there is something to pass.
There is a flame and a lamp. There must be! There cannot be no flame and no lamp! There must be something! This is how he ponders to himself.
Why is this so?
It's because for him the truth is yet to be known.
Then, a not yet liberated mind, reading what I already wrote, thinks to himself that the answer is nobody, thus comes to the conclusion that the answer is nobody and thus that there is nothing to pass ... no flame, no lamp.
There is no flame and no lamp he thinks to himself. There is nothing! There is no flame and no lamp! This is how he ponders to himself.
Then, at the time of his death, not knowing the cause of suffering, by volitional activities, he fabricates a flame of nothingness and a lamp of nothingness and destines himself to a future rebirth.
Why is this so?
It's because for him the truth is yet to be known.
In your comments you said that self dissolves after death in such a way that after death self does not remain. This cannot be, because the self in order to dissolve completely, needs to know the true nature of things. Without knowing the true nature of things, self cannot completely dissolve, thus it remains. This is key in understanding that after death some self remains and it does not completely dissolve.
The true nature of things is that there is no self. Thus, every self that exists (with or without body), is constantly and naturally returning to this true nature which is "no self", but instead of entering this natural state of "no self", the self arises again and again and again because of ignorance (not knowing the true nature of things). Or I can say it this way: every self, with the death of his physical body, is naturally returning to the true nature of things, which is Nibanna, but because of ignorance, the self raises into Samsara in a new body. In not liberated beings still existing in Samsara this ignorance exists due to kamma, on the other hand, in liberated beings still existing in Samsara, ignorance does not exist due to kamma but due to compassion. In reality, the ignorance in liberated beings still existing in Samsara does not exist any more, but they choose to be a part of this ignorance (Samsara) out of compassion for other beings.
Here is what happens after death ... or in the process of dying.
Self STARTS to gradually dissolve towards Nibanna ... it does not dissolve in an instant ... it just starts gradually dissolving ... firstly, it gets stripped from his body/form ... then it gets stripped from other things like memories, concepts, ideas, etc. ... it gets stripped from skandhas ... So after death, the self (the skhandhas) is gradually dissolving/stripping, but IT CAN NEVER COMPLETELY DISSOLVE (and thus enter Nibanna) IF IT DOES NOT KNOW THE TRUE NATURE OF THINGS (lets call this The Truth). The self can completely dissolve after death (enter Nibanna) only if The Truth is known to that self.
By not knowing The Truth, when after death the self is dissolving, volitional activities are made by the self (samkhara) which are influenced by kamma, and these same volitional activities stop the gradual dissolving process of the self towards Nibanna and start rebuilding the self, thus sprinkling the self in a new existence into Samsara either by birth, egg or apparition .... with or without body/form/senses, etc. ... it gets rebuilt into skandhas depending on his samkhara which depends on his kamma. The new existence of the rebuilt self is determined by the volitional activities of the self which are conditioned by kamma which is conditioned by the degree of not knowing The Truth.
This is the answer to your question about how does the craving survive after death.
To summarize: Craving survives after death because the self does not become completely extinct after death due to the fact that the self does not know The Truth. Self cannot become completely extinct/dissolved after death if he does not know The Truth. If self does not know The Truth, he will "fight" against his extinction/dissolution with craving, thus he will not reach extinction/dissolution (Nibanna) but will remain after death (stripped off from his skandhas) and be reborn again (Samsara) with newly fabricated skandhas arising because of craving.
In fact, death is not really death .... Just the self is stripped of some of his fabricated fabrications (skandhas) and this stripping of fabrications we call »death«. Then, because of his ignorance (not knowing The Truth), the self fabricates new fabrications (new skandhas) and this we call »birth«. In reality, the self never dies nor is born. »Death« and »birth« are just concepts (not truths) ... are labels that we give to a self when the self fabricates his body (skandhas) or his fabricated body gets destroyed (destroying of skandhas). Self is constantly in the cycle of death and rebirth, but we can't see it directly because of our kamma.
The claim that after death self dissolves/extincts is not correct, because for the complete extinction of the self, The Truth must be known to that self. The self cannot extinguish/dissolve if The Truth is not known to him ... or in other words: the self cannot extinguish/dissolve if ignorance is not removed by that self. This is what the Buddha taught.
Thus, the self remains after death and is reborn again. Only in a being who knows The Truth (who removed his ignorance) the self becomes completely extinct/dissolved after death, thus enters Nibanna and rebirth stops.
Upadana means fuel. This fuel is the self after death, stripped from his body/form, memories, ideas, etc. ... thus stripped from its skandhas. Kamma is the remaining fabricated fabrications of this stripped self. Not knowing The Truth, this self fabricates new fabrications based on his remaining kamma and in this way adds new kamma to his already remaining kamma. Fabricated fabrications can be body/form, memories, ideas, concepts, etc. ... thus adding of new kamma and rebuilding of skandhas.
In non liberated beings, after death and/or during life, because of avijja (ignorance), upadana is present. Upadana gives fuel to samkhara. Samkhara then rebuilds the self based on kamma and thus adds new kamma to his kamma. This self is not free from kamma. His kamma produces fruits, which due to the real nature of things (The Truth) always end in suffering, and he cannot stop this process due to ignorance.
In liberated beings who after reaching Nirvana decide to stay in their physical bodies, upadana is present because of compassion, not because of ignorance. Upadana gives fuel to samkhara. Samkhara then rebuilds the self based on kamma, but in reality this self is free from kamma. His kamma produces fruits only because he chooses to do so, out of compassion for other beings. He is free from suffering produced by fruits of his kamma because he knows the real nature of things (The Truth). In liberated beings, after death, there is no avijja (ignorance), thus upadana can exhausts completely. Without upadana there is no fuel for samkhara. Without samkhara, no self can be rebuilt, thus all the remaining kamma extinguishes and Nibanna is entered.
I think you need to study the second link in the Pratītyasamutpāda: samkhara. Also study tanha, upadana and bhava which are the main modus operandi of samkhara. Meditate and see with your own experience samkhara, tanha, upadana and bhava in action. Then everything will be clear.
You are very close. Do not get discouraged by my words. Do not believe my words. Meditate and verify it yourself with your own experience. Let these words be your guide in seeing The Truth in its entire beauty.
Metta to you and everybody