allow me to rephrase that:
Irrespective of being the intrinsic or extrinsic part of the body, the mind and memories would not survive death as an extension of a sober scientific perspective and hence memories won’t carry over which is needed for reincarnations.
That's one of the things that made humans so amazingly unique, we learned how to get around that road block to advancement, when hominids evolved language, art, writing.
Before taking the time to look up Richard Dawkins address, (I’m tempted to do it for you, but then you’ll never learn how to utilize google, ask and ye shall receive, even if you need to poke around before finding pay dirt.) refine your question.
How can reincarnations be explained from the scientific / rational perspective of Dr. Dawkins who says that there is no soul or consciousness?
Since, firstly I doubt Dawkins claims that consciousness doesn't exist. That would make an interesting thread in itself. But, if you misrepresent Dawkins from the get go, why do you think he'd bother to answer you?
It seems you haven’t even thought out the difference between “consciousness” and “soul” - since they are definitely not synonymous.
A little self-skepticism goes a long way. Give it a try.
As for the answer to your question as written.
The answer is a simple, there is no evidence of one’s consciousness being recycled into a new body. It makes no sense Lot’s and lots of anecdotes don’t make evidence.
Lots and lots of people believe in aliens from other worlds visiting us. Lots and lots of people believe Atlantis existed. Lots and lots of people believe that our amazingly complex global biosphere was created within a week’s time. Lots and lots of people believe in King Arthur and the holy grail. But, none have any actual evidence attached, only vague stories and self-certain personal faith. etc.
I’d suggest the bottomline is: Science is about stuff we can find evidence for. The rest is mind games within this tremendous Mindscape we possess and live within.