Well as to the original question...."What do you think? What transmigrates".
I will leave it to others of much greater knowledge than I, to do searches of sutra sutta and doctrine to estimate what approximates such a thing as explained in such a fashion.
In a personal sense and not to state it is even necessarily a Buddhist sense, though I may be able to "back it up" with certain references that are established within Buddhism. I do not actually know enough to state this is a "official Buddhist view". It seems so but I am not certain.
My take....We are essentially aware. To digress a bit "essentially" however does not imply inherantly existant. If no object presents we do not have a awareness present. This never however is the case for various reasons.
Our awareness has certain characteristics, present to cause what it is called our awareness. Ability to discriminate, ability to see sameness, ability to perceived energy movement dynamic, and.....specfic to this thing called rebirth, and what transmigrates....ability to retain things.This is a necessary process to know our environment and to know any object that may present in it. Awareness simply being to know of, as in knowing of the existance of a ball entering a room we may be in. Awareness in this context implies more cognicence, than that simplicity. Aspects of awareness enable us to "know" things. We know but relationship but that is another issue, to again digress.
So we are more than aware. Unfortunately we take these varying aspects of awareness(tools present to enable us to understand things) and misconstrue them. Like and dislike emenate from discrimination, combined with same perception, create a self concept(through ignorance) per example.
The rebirth or transmigration(actually nothing transmigrates we are the same essentially in death as in birth and all states inbetween) is a extension of the habitual tendency aspect of our awareness which is itself derived from the retentiveness quality aspect of our awareness, or cognitive process.
WE must retain knowledge to preform cognitive functions. Again that is a broader catagory but essentially this is the awareness that is spoken of...cognition. To fully know what a ball entering a room is, we must have a retentive aspect to our awareness, and from this stems habitual processes. Retention enables cognitive function to compare one object perceived to another, perhaps a object not currently being perceived.
So essentially by misconstruing awareness retentive aspect(the process how it works the mechanism) we create habit and thusly we create habitual tendency. Essentially with repetitive process and reinforcement(in this case self reinforcement of a mistaken kind) we create a thing we call self. Once we create self, a rebirth is a necessity and not a option. With creation of self....other is also created. Our birth death and all the rest(products of a misconstrued awareness) are a result of this creation. WE create self and other, and reinforce its existance with the production of a environment which affirms its existance. The particular rebirth is dependent upon what we consider to be necessary to reinforce self. We develope tendency(part of habitual formation) and from this tendency we create certain environments which serve in their complexity to in a final sense, reinfoce what we concieve of as our self.
So our tendency fits us to our particular rebirth. Self concept, necessitates it. SElf concept is the cause of a particuar rebirth.
So in essence, it is retentive aspect of consciousness from which rebirth stems.
So..."What do you think? What transmigrates"...is answered. I can elaborate on each essential point if requested.
A add on....
It is all total nonsense of course, these things are in the end simply how we understand, and self and other are all bogus garbage which deceives us into thinking this is how things are, when they are not. Only with true understanding, seeing things as they actually are through actually employing our awareness in their normal fashion, can we see through this pile of garbage which stretches to the sky and beyond. Right here and now is how things are when we die, no different in any aspect. Degree only.
Our death and rebirth is how we are. If we were not we would not participate is such painful things.
And not a shred of this changes the fact....without a object to preceive we do not perceive. That is the key to understanding our escape, not our awareness but our emptiness. So suchly we must travel in this particular direction if we want to end this thing that is, this delusion. The thing itself, our awareness quality our ability to do this thing called being aware, never ends when object is present to elicit its presence.
So there....enjoy my laypersons uneducated perhaps nonBuddhist view of this thing!!
Agree, or disagree, Buddhist or nonBuddhist, human or god, animal or demon, joker or thief, begger or king...I don't care....not a fig.
To add.....I will generally not debate this thing, as I consider it a personal view. And as personal view not better nor worse than any other personal view. I can elaborate if requested on some specific points, and can largly "back up" specific points from sources within Buddhism, if that is requested. I actually do not know what constitutes Buddhism at this place in time. Present day Buddhism remains largly unknown to me.
But I have answered the question, as I may, and has been requested.
"This order considers that progress can be achieved more rapidly during a single month of self-transformation through terrifying conditions in rough terrain and in "the abode of harmful forces" than through meditating for a period of three years in towns and monasteries"....Takpo Tashi Namgyal.