Daniel’s description is good. The following post also describes how a “lifestream” evolves without a “soul.”
“What Reincarnates? – Concept of a Lifestream”
Raj wrote: “The Buddha had encountered Dipankara Buddha thousands of eons ago. After that as a Boddhisatta, he took many forms (including many animal forms, as per Jataka tales), and on the night of his enlightenment he could see all these forms. So there was a continuity, it was the same individual, from the time he saw Dipankara Buddha, till the time he got enlightened under the Bodhi tree. What was it which remained same throughout that period?”
– NOTHING remained the same during that whole period.
– Each subsequent life arose based on the causes accumulated in the past AND conditions prevailing at the moment of grasping a new existence. This is what is explained in Paticca Samuppada, and this is hard to comprehend issue for many people.
Raj asked: “Is it the same Gandhabba undergoing slight modifications but still the same.”
– It is NOT the same gandhabba. Each time a new existence (human, Deva, animal, hell-being, etc) a BRAND NEW gandhabba is created by kammic energy.
– This is essentially the difference between Vedi teachings and Buddha Dhamma.
– By the way, Vedic teachings originated with Buddha Kassapa who lived before Buddha Gotama. That is why there are some similarities. Vedas took the concept of gandhabba and turned it into an “indestructible atman/soul”. I have discussed this briefly in some posts. See, for example, #8 in “Arōgyā Paramā Lābhā..”
I know it is very hard to get rid of the perception of a “me”. In fact, even a Sotapanna has that PERCEPTION, but a Sotapanna has seen that there is NOTHING that migrates through adjacent lives. The perception of a “me” goes away only at the Anagami stage. Then there is still a trace of “me” left and that goes away completely at the Arahant stage.