Blavatsky
Original sense
The reincarnating entity is the higher Ego (Manas), not the personality. Personal memory usually dies with the body. Lives are separated by long Devachanic intervals, not by a quick bounce. She rejected the popular idea that humans reincarnate as animals (a misreading of transmigration), and she rejected ‘I remember being Cleopatra’ as, usually, astral-light romance.
Reincarnation means that this Ego will be furnished with a new body, a new brain, and a new memory.
Karma and reincarnation are, for her, a single law. She also claimed the number of human monads is finite for this cycle — hence old karmic debts from Atlantis still being paid.