Personally, I like the idea that your mind isn't moving anywhere at all...
...it's this damned stream of visitors who keep showing up, lifetime after lifetime,
each one claiming to be the real "me".
My understanding of this (and it is only my understanding) is very simple.
Mental patterns tend to replicate themselves pretty closely.
The way the Buddha expressed it, like the wheel of the cart that follows the ox who pulls it.
That is why it feels like you are the same person from moment to moment.
But they do not replicate themselves exactly. Everything changes.
This is why you are not exactly the same person from moment to moment.
If you are driving a truck that is carrying a pile of wooden boards, and the truck turns over near a river,
the pile is dispersed, and all the boards go into the river,
float down stream, and wash up in different places.
But some of those boards also wash up in the same place, in another grouping.
Other boards may wash up somewhere else.
These boards are like the aggregate causes
of cognition (but not actual cognition itself) ,
and the truck turning over is like the body when an ordinary person dies.
Suppose another person finds a lot of those boards and puts them into his truck, into a new "pile".
Maybe that wood will be burned up, or made into a table or something.
This is like the causes of cognition moving from one body to another.
The cognition, the 'consciousness' did not move,
but what moves are the causes
of cognition, or of consciousness.
Suppose another truck turns over and the wood boards falls out into the river,
but these wood boards happen to have first been made into a boat.
Then when the boards (boat) falls into the water, it retains its shape.
All the boards float down stream, but having first been assembled into a boat,
they arrive downstream as essentially the same boat.
This is like what happens to realized beings, such as some lamas, when they die,
because they have organized their minds into vessels rather than just a loose stack of mental aggregates.
It's not a physical thing that moves on. There is not some cartoon thought balloon floating invisibly through space,
out of a corpse one place and into a fetus somewhere else.