Reincarnation of Elijah

Despite all the modifications of the intial texts, still have been preserved some references about reincarnation, as put in the mouth of Jesus concerning of Elijahs reincarnation as John the Baptist, as given in:
Matthew [11,12-14],
12 From the days of John the Baptist until now, the kingdom of heaven has been subjected to violence,[a] and violent people have been raiding it.
13 For all the Prophets and the Law prophesied until John.
14 And if you are willing to accept it, he is the Elijah who was to come. 

Matthew 17, 10-13
10 The disciples asked him, "Why then do the teachers of the law say that Elijah must come first?"
11 Jesus replied, "To be sure, Elijah comes and will restore all things.
12 But I tell you, Elijah has already come, and they did not recognize him, ...."
13 Then the disciples understood that he was talking to them about John the Baptist.

Christian objection: 
One first has to die in order to be reincarnated.
Answer:
Reicarnation means to change a body and the consciousness attached to that body with another body which may have parts of the previous consciousness.

As recorded in the Bible Elijah was raised to Heaven, and although not written he got a heavenly body, not the human body as he had on earh.
The fact that Elijah was previously a human is recorded in Jacobs 5:17
"Elijah was a man with a nature like ours, and he prayed earnestly that it might not rain..."

As Elijah reincarnated as "John the Baptist " it is not said, he had falled from Heaven,
but that he was born in a human body, having as parents according to Luke 1:13 some Zechariah Elizabeth.
Luke 1:13 "Do not be afraid, Zechariah; your prayer has been heard. Your wife Elizabeth will bear you a son, and you are to call him John."

According to the Knowledge about the Migration of the souls, they migrate on all the planetary systems, from the heavenly ones to the hellisch one, according their deeds and attaind consciousness until they attain a transcendental (spiritual) consciousness, see The first part of Creation.