In order to try and relate how the process of rebirth is falsifiable and therefore try and relate the concept to science, I will have to establish a number of ideas/theories before I can directly try and relate it to science, so I'll have a bunch of definitions or excerpts from sutras and then I'll try and relate it all together, so the beginning of my question may be just listing a bunch of theories which will later be brought together.
The process of "rebirth," as I understand it, doesn't allow for one to remember a past life or past life experiences. It is important to make the distinction between the physical Buddha (Shakyamuni himself) and, what is in essence, the "life force" of the Buddha. Shakyamuni himself wasn't able to see his past lives or remember past experiences from before he was born. What he was able to do, and this is what his fifth transcendental power of being able to see past lives is, is understand his "life force" and perceive the nature of his life, which he had in his previous lives. It is from this concept that karma/karmic chains and other such related ideas come from. To support this affirmation, in the Lotus Sutra, chapter 16 (The Life-span of the Thus-come One), when Shakyamuni Buddha is addressing an assembly of nearly all of his followers, he says:
"You must listen carefully and hear of the Thus Come One’s secret and his transcendental powers. In all the worlds the heavenly and human beings and asuras all believe that the present Shakyamuni Buddha, after leaving the palace of the Shakyas, seated himself in the place of
practice not far from the city of Gaya and there attained annuttara-samyak-sambodhi. But good men, it has been immeasurable, boundless hundreds, thousands, ten thousands, millions of nayutas of kalpas since
I in fact attained Buddhahood.... in truth the time since I attained Buddhahood is extremely long (1)".
Shakyamuni here is saying that there was no start to his "existence" or life force in it of itself. Basically, he is saying that his, and everyone else's life force has existed from "the infinite past" and will continue to exist throughout. What is important to know is that one's life force isn't life in the standard sense, but is intrinsically existing as a function of the universe regardless of whether there is life. Life is more the manifestation of one's life force.
Also in the Lotus Sutra, in chapter 14 (Peaceful Practices), Shakyamuni states,
"All phenomena are empty, without being, without any constant abiding, without arising or extinction.... phenomena exist, do not exist, are real, are not real, are born, are not born.... Look upon all phenomena as having no existence, like empty space, as without firmness or hardness, not born, not emerging, not moving, not regressing, constantly abiding in a single aspect (1)".
Here, Shakyamuni is saying that the true aspect of all phenomena is that it has no existence. Another way to say this is that causes don't have intrinsic implications; only effects do. It also is to say that, whatever happens, that doesn't mean anything other than that that happened; how one reacts is all that matters. Here one can see how Shakyamuni's statments of the true aspect of all phenomena can be easily applied to our day-to-day lives, so the next logical step is to have a system of, in essence, figuring out which different effects one can create from any phenomena. This leads us into the next philosophical idea.
The 3000 Realms is an philosophical theory developed by T'ien-T'ai, a Chinese buddhist philosopher. It is to say that the, "entire phenomenal world exists within a single moment of life (2)". The realms themselves are used as a way to "categorize" the phenomenal world. Below is a chart explaining how he came up with 3000 realms:
The Ten Worlds (or Ten Spiritual Realms) are basically thought to be the full spectrum of one's life condition, which comes from ones life force (i.e. how one feels). I'm not entirely sure where the concept of the ten worlds comes from, although I believe it is a relatively universall concept, at least within Mahayana Buddhism.
In short, however one feels can be represented as one of the "Ten Worlds", or "Spiritual Realms". The part of the chart which lists it as "The Ten Worlds and Their Mutual Possession" is referring to the fact that T'ien-T'ai considers the ten worlds to exist within each other. That is to mean that, even in the realm of hell, one can exist in a realm of buddhahood. An example from history of this concept would be of Martin Luther-King Jr.. He led the civil rights movement in America for some time and did so peacefully and, as far as I know, ethically too. One may say that he often times existed in one of the upper worlds like Buddhahood or Bodhisattva (I don't know the Ten Worlds well enough to be able to say with much confidence which particular world he was in while he was working for Civil Rights). Anyhow, while he was doing that though, he was angry because of the injustice of how he and other people were being treated and that motivated him. However, the important distinction to be made is that, although he was angry, he didn't act violently, he still thought and acted like one who was in one of the higher worlds would act. The inclusive world which he often dwelled in was Anger, even though he was in one of the higher worlds nonetheless.
Another important aspect of the Ten Worlds is that the entirety of them exist intrinsically within us at all times and one can, moment-to-moment, move from one world to the next- either up or down. That is to say that, at all times, for all people without exception, the World of Hell exists within us, but so does Buddhahood.
The next part of The 3000 Realms is The Ten Factors. This theory too comes from the Lotus Sutra in the second chapter (Expedient Means) and is expounded by Shakyamuni to Shariputra and goes as follows:
"Shariputra, to sum it up: The buddhas have fully realized the law that is limitless, boundless, never attained before.... The true aspect of all phenomena can only be understood and shared between buddhas. This reality consists of appearance, nature, entity, power, influence, internal cause, external cause, latent effect, manifest effect, and their consistency from beginning to end (1)".
I personally have a hard time explaining The Ten Factors and so I'll also add a link that describes The Ten Factors (3) more clearly. To the best of my understanding, The Ten Worlds are how we feel emotionally, and the Ten Factors are how we project ourselves upon the world and how these two concepts interact with each other.
The final part of The Three Thousand Realms is called, "The Three Realms". I know the least about this part of the Realms, and so my explanation for it probably wouldn't be complete, but there are many websites which I think would be able to better explain them. Everything that I would be able to say about them is addressed in the image I added above, so I'll stop here.
What is important to know about The 3000 Realms as a whole is that they describe all of the possible effects within a single moment of life. It also means that everything is subjective - in terms of point-of-view - and that all things are equal.
The reason that I have included the concept of The 3000 realms is because it shows that not just Shakyamuni has a life force, and also that one's enlightenment (world of Buddhahood) intrinsically exists within one's self (as well as every other realm).
All perspectives are equal because anything that we perceive is a perception of a phenomena and all phenomena are "empty", as stated above, thus meaning that everyone's perceptions are equal.
This would mean that, if one can prove that all perspectives are equal, one can prove the principle of rebirth to be falsifiable because: If all perspectives are equal, because all phenomena are empty because there is nothing to perceive, all that is left to exist is our life force. Because our life force isn't constrained by any phenomena (e.g. birth and death), it exists for eternity neither beginning nor ending. One however, can not say that life simply doesn't exist, because as Descartes said, "I think therefore I am." This would leave the only option left being that "rebirth" exists, because our physical selves are bound by physical limitations and phenomena, things will be born and will die. However, since our life force is eternal, and not really "alive" in the sense that living is usually implied with, when we are physically living we embody that life force just as how, when there is matter, there will be gravity. Our life force is eternal, however, we, as living beings, are not, and so we will forever be born and die, but whenever we are born, we embody our life force. This is what reincarnation is.
What is left to do is then prove that all perspectives are equal. For that one must look at Einstein's Theory of Relativity. I don't understand this concept in its entirety, but I do know that it says that, if there is a person who isn't moving, time will be faster for him or her than it is for someone who is moving. This has been scientifically proven to happen. Time wouldn't change for certain people (or anything for that matter), if someone's perspective could be deemed invalid. It is because everyone's perspective is equal, that time itself changes when something moves.
I hope this answers your question. I know that some of my explanation may be a bit murky or unclear, so I would be happy to try and clear up anything.
1) The Lotus Sutra and Its Opening and Closing Sutras. Tokyo: Soka Gakkai, 2009. 57-267. Print.
2) "Three Thousand Realms in a Single Moment of Life." Soka Gakkai International Buddhism in Action for Peace. Soka Gakkai International, Apr. 2012. Web. 6 Aug. 2016. http://www.sgi.org/about-us/buddhism-in-daily-life/three-thousand-realms-in-a-single-moment-of-life.html.
3)"Ten Factors." Soka Gakkai International Buddhism in Action for Peace. Soka Gakkai International, Oct. 2000. Web. 6 Aug. 2016. http://www.sgi.org/about-us/buddhism-in-daily-life/ten-factors.html.