I think its a big difference between what adults claim and what children claim. Adults are surely influenced by experiences they made and believes they grew up with. Little children are not. If a little child tells you things it hasnt heard or learned anout, out of the blue, thats more believable imho, then you or i claiming something. The brain and it's functions are pretty much not well charted. Neuroscience can not answer many questions but has a lot of theories there to be proven. This life is this life and what comes after that is something completely different. I would see it as a new start without burdons to be carried around. But after all, who cpuld possibly know.
Neuroscience has already answered that once our brain stops functioning, we are gone unless there's new evidence that you have that it isn't the case, that there is a part of us that lives on again because the self as you and I are cannot survive the destruction of the brain.
The problem with children when it comes to things like this is that children have very active imaginations. They are unreliable narrators. There's also the problem that our memories are physical so how would a child's brain that isn't even functioning at full capacity even be able to make coherent sense of memories that aren't even theirs but it's still theirs but from another life lived.
There's just too many problems with reincarnation for me to warrant taking it seriously and many have already convinced themselves that this is what will happen after death.
Philosophically, if you have no memory or connection to a past life, then how is that you? Personality and self are physical constructs of the brain, as evidenced by the myriad cases where damage to the brain changed someone's whole personality.
It requires there to be some sort of soul that carries a record of your life, that is measured after your death and then reassigned to some new lifeform to try again. All of which is 'not even wrong'. There's no possible way to have any evidence against that, since there's never been any evidence for it.
Until a mechanism by which it could function can be proposed and tested, there is no reason to treat it as a coherent or reasonable claim.
But I get it, some of you want a chance to experience the things you never stood a chance at in this life, to not live with the same trauma or illness you have right now, to be beautiful, to experience love, be healthier or successful in another life, to live out the opportunities you don't have now, to no longer be burdened by the life you live, that you want to get away from. Problem is, living in just another time or part of the world doesn't guarantee you will be happier, you may even be worse off, still dealing with the same existential crisis that comes with being a human. Whose to say even being born as someone else will be everything you imagine it to be, without considering the trade-offs. I don't find fault in that like people out there who believe they'll live for eternity worshipping whatever God they believe and that they'll be happy forever, it's always about getting away from the human condition you are desperately still attached to.