The question here is a bit difficult to frame sensibly, partly because it is unclear what should be taken to mean when someone claims that one prophetic figure is (in some sense) the incarnation of an earlier prophetic figure. It is especially unclear when operating within a worldview that denies reincarnation in favor of another view.
The problem here is both shallower and deeper than it might seem. Let's start in the shallow end, with the sayings of Jesus from two of the synoptic gospels along with angelic testimony from the third:
Mark 9:10-13 (KJV)
And they kept that saying with themselves, questioning one with another what the rising from the dead should mean. And they asked him, saying, Why say the scribes that Elias must first come? And he answered and told them, Elias verily cometh first, and restoreth all things; and how it is written of the Son of man, that he must suffer many things, and be set at nought. But I say unto you, That Elias is indeed come, and they have done unto him whatsoever they listed, as it is written of him.
Matt 11:10-14 (KJV)
For this is he, of whom it is written, Behold, I send my messenger before thy face, which shall prepare thy way before thee. Verily I say unto you, Among them that are born of women there hath not risen a greater than John the Baptist: notwithstanding he that is least in the kingdom of heaven is greater than he. And from the days of John the Baptist until now the kingdom of heaven suffereth violence, and the violent take it by force. For all the prophets and the law prophesied until John. And if ye will receive it, this is Elias, which was for to come.
Luke 1:13-17 (KJV)
But the angel said unto him, Fear not, Zacharias: for thy prayer is heard; and thy wife Elisabeth shall bear thee a son, and thou shalt call his name John. And thou shalt have joy and gladness; and many shall rejoice at his birth. For he shall be great in the sight of the Lord, and shall drink neither wine nor strong drink; and he shall be filled with the Holy Ghost, even from his mother's womb. And many of the children of Israel shall he turn to the Lord their God. And he shall go before him in the spirit and power of Elias, to turn the hearts of the fathers to the children, and the disobedient to the wisdom of the just; to make ready a people prepared for the Lord.
Taking all the synoptics together, we are lead to believe that John the Baptist is to be identified (in some missional or spiritual sense) with Elijah the Prophet, especially respecting the fulfillment of a prophecy found in the final chapter of the Old Testament. Seems clear enough, that is, until we get to the fourth gospel.
John 1:19-25 (NASB)
This is the testimony of John, when the Jews sent to him priests and Levites from Jerusalem to ask him, “Who are you?” And he confessed and did not deny, but confessed, “I am not the Christ.” They asked him, “What then? Are you Elijah?” And he said, “I am not.” “Are you the Prophet?” And he answered, “No.” Then they said to him, “Who are you, so that we may give an answer to those who sent us? What do you say about yourself?” He said, “I am A VOICE OF ONE CRYING IN THE WILDERNESS, ‘MAKE STRAIGHT THE WAY OF THE LORD,’ as Isaiah the prophet said.” Now they had been sent from the Pharisees. They asked him, and said to him, “Why then are you baptizing, if you are not the Christ, nor Elijah, nor the Prophet?”
Here we have John the Baptist flat-out denying that he is to be considered either Christ or Elijah in the sense those terms were taken to mean by his hearers. Instead of identifying himself with the prophecy of Malachi 4:5-6 he identifies himself with another prophecy from another prophet. How is this conflict between the synoptics and the Johannine narrative to be resolved?
A favorite apologetical trick consists of a bizarre reversal of the fallacy of equivocation, where the apologist explicitly claims that the same term (or phrase) takes on completely different meanings in similar biblical contexts. Here, the apologetical equivocation takes the following form:
1) "He is Elijah" means one thing in the synoptics
2) "He is Elijah" means something else in gJohn
:. He is both Elijah (sense 1) and not Elijah (sense 2)
This could be made to work, of course, if a good contextual or cultural reason is given (beyond special pleading for the sake of forced harmonization) to believe that we really should take the same phrase in two different senses. In this case, I've yet to see any good reasons given to warrant such semantical sleight of hand.