These verses connect a single doctrine. That doctrine is at the heart of "turning of the hearts of the children to the fathers," which is the result of any restoration of the Gospel. The definition of "children of the prophets" is that one has accepted, believed, and followed the Lord's true messengers. They become children of Abraham and receive priestly authority sealing them into the family of God, joining the "fathers." From the time of Abraham until today, all who are redeemed have become a part of his household.
The phrase "turning the hearts of the children to the fathers" is a reference to the restoration of sealing authority, allowing a connection between man living on the earth, and the fathers (Abraham, Isaac and Jacob). In this dispensation, that restoration occurred when Joseph Smith was given the sealing authority and priesthood whereby he could ask and receive answers. (D&C 132: 45-47.) As discussed earlier, this was sometime between 1829 and 1833, though I think it was more likely the earlier date as I have explained. Coincident with receiving this authority, Joseph's calling and election was made sure. (D&C 132: 49.) I have explained this in Beloved Enos. This priesthood, having the hearts of the recipients turned to the fathers, and the promise of exaltation, are interconnected.
Abraham not only held this authority, but received the promise that all who received the Gospel after him would become his descendants. From the time of Abraham to the present, every saved soul has had their heart turned to him, become his son or daughter, and receive that same priesthood. (Abraham 2: 10-11.) When Joseph received this, he was not merely sealed up to eternal life, but he became part of the family of Abraham. If you remember the diagram of the celestial kingdom referred to earlier on this blog, you know Joseph became one of those who was grafted into the family tree, and would then in turn preside over others who were sealed up to eternal life thereafter.
The sealing authority used by Joseph in December, 1832, was to seal others up to eternal life. (See D&C 88: 2-4.) This promise had been previously conferred upon Joseph in that portion of Section 132 referred to above. In fact, Joseph's use of that authority in December, 1832 on behalf of others is evidence that the promise to him recorded in Section 132 was necessarily received earlier than December, 1832. If it had not been received earlier, there would have been no need to make the statement in D&C 132: 49 to Joseph, because of what is in Section 88. Why tell Joseph his calling and election was sure in 1843 if it had happened already in 1832? This is another reason you can know Section 132, although recorded in 1843, was in fact a revelation received by Joseph much earlier. It was reduced to writing in 1843 at Hyrum's request.
The reference to "turning the hearts of the fathers to the children" made by Elijah was not because Elijah conferred those keys upon Joseph in the Kirtland Temple (D&C 110: 16), for they arrived years earlier than 1836. Elijah was confirming that the keys were now all returned so the hearts of the children could turn to the fathers, and in turn the father's hearts to the children. (D&C 110: 15.) And, so as to signify he was a true messenger, Elijah also showed a sign by his hand to Joseph whereby Joseph could recognize a true messenger. (D&C 2: 1.)
In Christ's statement to the Nephite audience, He confirmed that they were "the children of the prophets" because they followed the prophets' teachings. Therefore, because of their obedience they were "of the house of Israel" and had realized that status because "of the covenant which the Father made with your fathers." That covenant was given "unto Abraham" promising to Abraham: "And in thy seed shall all the kindreds of the earth be blessed." All those after the day of Abraham who received this priesthood and sealing would become the seed of Abraham. They become heirs of the promise, and children of Abraham. They are sealed up to eternal life and therefore their hearts have turned to the fathers.
Christ was sent to these Nephites because, "The Father having raised me up unto you first, and sent me to bless you in turning away every one of you from his iniquities; and this because ye are the children of the covenant." Realizing the promises, and being visited by the Lord are also connected. When enough are ready to enter into this order, have their hearts turned to the fathers, receive the covenant, then the Lord will bring again Zion.