Most Christian teachers teach that there is no such thing as reincarnation (as in Hinduism and Buddhism).
But the Bible says that the spirit of one man may come into another.
“Yahweh came down in the cloud, and spoke to him, and took of the Spirit that was on him, and put it on the seventy elders: and it happened that when the Spirit rested on them, they prophesied, but they did so no more.” (Num. 11:25)
So the spirit of Moses came upon 70 other people. It can be called reincarnation. Likewise the spirit of Elijah has come onto Elisha (2Kin. 2).
This sometimes may take the form of a classic “reincarnation” where one person remembers the life of another dead person through the spirit.
Jesus said speaking of John the Baptizer, “If you are willing to receive it, this is Elijah, who is to come.” (Mt. 11:14)
This means that John was a reincarnation of Elijah for those people willing to believe that he was Elijah, but not for those unwilling to accept the spirit of Elijah. That is, the first category of people speaking with John would have heard Elijah, but the second group would not.
“But I tell you that Elijah has come already, and they didn’t recognize him, but did to him whatever they wanted to… Then the disciples understood that he spoke to them of John the Baptizer.” (Mt. 17:12)
“He said to them, ‘Elijah indeed comes first, and restores all things. How is it written about the Son of Man, that he should suffer many things and be despised? 13 But I tell you that Elijah has come, and they have also done to him whatever they wanted to, even as it is written about him.’” (Mrk. 9:12-13)
“He will go before him in the spirit and power of Elijah…” (Luk. 1:17)