Where is this 'promise' found?
Well, considering how and why we have the biblical records we do, it’s because God has always commanded his children to write their histories, it makes perfect sense that his other tribes would also do the same over the last few millenia of time. Why would it be any different.
BTW, We have this “promise” recorded in the Book of Mormon:
2 Nephi 29:
10 Wherefore, because that ye have a Bible ye need not suppose that it contains all my words; neither need ye suppose that I have not caused more to be written.
11 For I command all men, both in the east and in the west, and in the north, and in the south, and in the islands of the sea, that they shall write the words which I speak unto them; for out of the books which shall be written I will judge the world, every man according to their works, according to that which is written.
12 For behold, I shall speak unto the Jews and they shall write it; and I shall also speak unto the Nephites and they shall write it; and I shall also speak unto the other tribes of the house of Israel, which I have led away, and they shall write it; and I shall also speak unto all nations of the earth and they shall write it.
13 And it shall come to pass that the Jews shall have the words of the Nephites, and the Nephites shall have the words of the Jews; and the Nephites and the Jews shall have the words of the lost tribes of Israel; and the lost tribes of Israel shall have the words of the Nephites and the Jews.
14 And it shall come to pass that my people, which are of the house of Israel, shall be gathered home unto the lands of their possessions; and my word also shall be gathered in one. And I will show unto them that fight against my word and against my people, who are of the house of Israel, that I am God, and that I covenanted with Abraham that I would remember his seed forever.
Further, our belief is that the Lord himself ministered to the lost tribes, for they are not lost unto God. The lost ten tribes will produce the third witness that Jesus is the Christ. Their record will contain the story of Christ’s visit to them after his ascension and visit to the Nephites:
3 Ne 17:4 But now I go unto the Father, and also to show myself unto the lost tribes of Israel, for they are not lost unto the father, for he knoweth wither he hath taken them.
One last reference from the New Testament:
2 Corinthians 13:1 In the mouth of two or three witnesses shall every word be established.