The Indian girl suddenly starts to describe the medieval English castle, and to name  the ones who in certain time was dwelling in it. Her recollections prove to be in full accordance with chronicles of this family. Thus, she is this Englishwoman (or the Englishman) in a new embodiment?
  Before the Tibetan boy they display the different things among which he unmistakably identifies which was belonged to late dalaj-llama. Thus he is the same unique and eternal the dalaj-llama which lives now in the new body? But what is reincarnation?

For the Indian and Tibetan consciousness these conclusions are self-evident and do not cause any doubts. But the person of Judaic, Muslim or Christian culture for whom the idea of reincarnation should seem absurdity, what should he think about this facts? We do not speak about scientifically-materialistic consciousness, because for it the only way to save itself from frustration – is to deny categorically the reliability of the similar facts.

  Attempts to alter the Bible doctrine on east manner seem quite helpless. Pair of citations from the Bible used in this goal, speak absolutely about other. So, the Lord proclaims to prophet Jeremiah:

"Before I formed thee in the belly I knew thee; and before thou camest forth out of the womb I sanctified thee, and I ordained thee a prophet unto the nations" (Jer. 1:5).

Here nothing to do with "reincarnation", but the God ability to foresee does not cause doubts for the believing person.

In the book of prophet Malachi there are God's words: "Behold, I will send you Elijah the prophet before the coming of the great and dreadful day of the LORD" (Malachi 4:5).

But all interpreters understood this text as intimation that Elijah remains live, to preach the truth (together with patriarch Enoch) in hard times of Antichrist which will kill both of them.

As well Jesus words about John the Baptist: "And if ye will receive it, this is Elias, which was for to come. He that hath ears to hear, let him hear" (Мt. 11:14) mean only that John in the first Coming of the Christ plays the same role which Elijah should fulfill in the second.
  And all picture of the world, all ethics of monotheism based on uniqueness and absolute value of human life would be completely destroyed by an assumption of idea of reincarnation.

However, attempts to connect the unjoinable were undertaken repeatedly.

Such experience of large scale was shown by the Russian thinker, poet and mystic Daniel Andreev (+1959). In the monography «The Rose of the World» he connects the Christian mode of feeling, the orthodox ethics and  aesthetics with rationalistic conception based on idea of "monad", which have passing a way of spiritual growing up through a long line of embodiments. Deep pathos of his theory was connected with the vital task of epoch: to defend autonomy of the person in the front of the overwhelming totalitarian violence – therefore for him is so important the image of indestructible monad. In this case it is possible to speak about «noble inconsistency», about unwillingness to make to the end all conclusions from accepted picture of the world. See English  text  http://www.roseofworld.org/

Inasmuch as Indian and Tibetan experience of "former lives» cannot be rejected, it requires the convincing explanation acceptable from point of view of Bible and Koran.

I see here only one decision.

Except the man, the God has created the bodiless spirits (angels) possessing certain freedom in their behaviour. Some category of these spirits had chosen a special way of existence: to take up their abode in bodies of animals, and then of the man. The Almighty God does not forbid similar actions till certain time though, obviously, He does not approve them.

In the same body arises some kind of "symbiosis" of incorporeal spirit with human soul. It is impossible to name this symbiosis peaceful because everyone of two active and conscious beginnings aspires to subordinate another and to become the unique bearer of human "I".

The position in this struggle defines the essence of religious culture. If the monotheism stands entirely on the part of the man and tries to expel illegally installed spirit, the east religions identify the person with this spirit – with all mental, psychological and moral consequences.

Therefore, from point of monotheism, we can tell: the religion of reincarnation is not human religion, but religion of these installed spirits. It is not our religion, it is their religion. And all problems connected with reincarnation: karma, sansara, release - these are not our problems, these are their problems.

Nevertheless, we should reflect: for what purpose the Lord supposes such scale spiritual experience? Besides it is impossible to deny: east wisdom, despite its fundamental error, brings for the person many the most valuable fruits.

Perhaps, symbiosis of angels with the man is forseen by the Divine design, but such union will become possible and admissible only when the person will reach the necessary level of a spiritual maturity?

And those angels who have kept fidelity to their Creator, do not try to advance events by self own will, but use all reasonable efforts to help the man to reach, at last, this necessary maturity.

And though we give our own answer on question what is reincarnation, one of steps to this matutity is the deep, sincere, sober comprehension of immense  experience of east mysticism.