Peter wrote:I don't recall the Buddha saying anything about the mechanics of it other than saying that craving makes it happen. Without craving, it doesn't happen.
I Agree. As I know the Buddha never taught the mechanics of rebirth. He said as long as there are ignorance and craving there is rebirth. There are some suttas about Kamma and how the beings will reappear depending to their thoughts and doings. And knowing the legality of Kamma is imho the knowing of the mechanisms of rebirth. But the Buddha warned us against speculation about Kamma in the Acintita Sutta:
"There are these four unconjecturables that are not to be conjectured about, that would bring madness & vexation to anyone who conjectured about them. Which four?
"The [precise working out of the] results of kamma...
In the same direction leads Samyutta Nikaya 56.31:
Once the Blessed One was staying at Kosambi in the simsapa forest. Then, picking up a few simsapa leaves with his hand, he asked the monks, "What do you think, monks: Which are more numerous, the few simsapa leaves in my hand or those overhead in the simsapa forest?"
"The leaves in the hand of the Blessed One are few in number, lord. Those overhead in the simsapa forest are more numerous."
"In the same way, monks, those things that I have known with direct knowledge but have not taught are far more numerous [than what I have taught]. And why haven't I taught them? Because they are not connected with the goal, do not relate to the rudiments of the holy life, and do not lead to disenchantment, to dispassion, to cessation, to calm, to direct knowledge, to self-awakening, to Unbinding. That is why I have not taught them.
"And what have I taught? 'This is stress... This is the origination of stress... This is the cessation of stress... This is the path of practice leading to the cessation of stress': This is what I have taught. And why have I taught these things? Because they are connected with the goal, relate to the rudiments of the holy life, and lead to disenchantment, to dispassion, to cessation, to calm, to direct knowledge, to self-awakening, to Unbinding. This is why I have taught them.
"Therefore your duty is the contemplation, 'This is stress... This is the origination of stress... This is the cessation of stress.' Your duty is the contemplation, 'This is the path of practice leading to the cessation of stress.'"
So the Buddha thought that it is not useful for salvation to know the exact mechanism of kamma and rebirth. We have to trust the Buddha that there is rebirth as long as there is ignorance and craving. There is no necessity to know the mechanics of it. I think it could lead us in the wrong direction. I'm sure that not even every Arahant knows of past existences (but maybe some). The Arahant knows 'birth is ended'.