What aspects of mind transmigrate during rebirth?
I'm not sure what you mean here, Tad, so I will attempt to express a response based on what I believe you mean, and please correct me, if I missed the point you were trying to make.
So, rebirth is often referred to as transmigration of the soul, and I agree with this basic premise, but then your question becomes relevant when we try to understand the difference between soul and identity. I think it was Freud who published a book called "Ego and the Id." I read it decades ago, and don't recall much of it, so I will just express my thoughts at present on this topic.
The term 'ego' is derived from the Greek term for the soul, whereas, cultural concepts of ego are more bound up in identity, which represents cultural confusion over these topics. The soul is pure, unidentified, consciousness and most of us have little contact with it, because we are so bound up in identity.
Identity is associated with the physical body and its history, which is its memory. As we develop as contemplatives we move through 8 stages of ecstasy which represents becoming aware of and identifying with the soul, which requires relinquishing identity, and is represented by the second stage of ecstasy, or second jhana where the mind becomes still. As we traverse the 4 jhanas we shed the layers of identity and become more just pure consciousness, which is soul. When we move to the 5th ecstasy we have shed the body and are only soul/consciousness.
We should recognize at this point that another aspect of what transmigrates from lifetime to lifetime are fragments of identity based upon experience in the material world, which more often than not are traumatic experiences that for some reason make it into the next lifetime and are represented by deep fears and attachments without a history to support them, until we begin to recall previous lifetimes when we find out the source of these attachments and traumas.