One thing we know for sure from studies of cases of rebirth is that they transfer information, from the dead previous body to the next person being born.
Of course most of us do not remember our previous life, thus the main lack of evidence that inspires lack of belief in this phenomena. Regardless, there are some people who spontaneously remember their past lives (and found facts in the present that agrees with the memory) and it is possible to remember past life information via meditation practice, so for the purpose of this thought experiment, rebirth is equal to information transfer or at least the potential for it.
Now consider the scenario that humans as a species survive the ongoing climate change, possible nuclear war and colonise space without being exterminated by hostile aliens and not creating enemies by being hostile to aliens. Humans would spread out not only throughout the whole galaxy, which is merely 100,000 light years across, but the whole universe, in particular, consider galaxies that are about 1 billion light years apart from each other. Assuming faster than light travel and time travel via general relativistic time machines are prohibited or plain impossible, there would be no practical means of communication between the two galaxies because by the time a laser communication device reach from one galaxy to another, one billion years might had wiped out all other traces of civilisation from the sender.
However, consider this: what if one person from this galaxy gets reborn in another one about one billion light years away? According to the Theravada Buddhist tradition, the rebirth is instantaneous (from the point of view of the consciousness, the death consciousness is immediately followed by a rebirth consciousness). Bringing this back to Physics, instantaneous between two bodies means two events (death and rebirth) happening at different place at the same time, or simultaneous.
The problem is, there is no such thing as absolute simultaneity. Special Relativity (SR) threw that out long time ago, the plane of simultaneous events changes according to your velocity.
Here we shall make a short detour to address if SR really applies in intergalactic distances. SR is just a local flat approximation of General Relativity (GR), the theory that made Einstein famous. For universal scales of intergalactic distances, we should use GR. Yet, now we know that in the large scale of the universe, it is very near to being a flat universe, or that the mass distribution is uniform enough that general relativity does not matter (much) in the large scale. That means special relativity could be applied across intergalactic distances.
Back to planes of simultaneity. There would be no such thing as that plane in the universal scale. Thus if rebirth is instantaneous from the outside, this begs the question: instantaneous to who?
If we can solve this problem, we might have information time travel, or even time travel itself!
One easy way out is to refer to the one of the cases in Francis Story's book: Rebirth: Doctrine and Case Studies.
In this case, the person who died had