" In Buddhism, the phenomenon of rebirth is linked to the continuity of the flow of consciousness, which is not restricted to our present life."
" When they hear the word reincarnation, many people think it means that when the body dies, part of the self remains and retuns in a new body. According to Buddhism, however , the "self" is but an illusion and can not therefore reincarnate itself.
The relatiosnhsip between the continuum of consciousness and the self is the same as that between a river and its name: it is simply a conceptual designation with no existence of its own."
source: Buddhist Himalayas, Matthie Ricard & Olivier & Danielle Föllmi