The last of the seven Durins appears once in Appendix A to The Lord of the Rings, as an undated descendant of Thorin III Stonehelm. As such, we know almost nothing of him apart from the bare fact of his existence.
In volume 12 of The History of Middle-earth (The Peoples of Middle-earth), though, we have access to the proofs of Appendix A, and a little more of Durin's life is revealed. His birth, it seems, was prophesied at the Battle of Five Armies, and he was born several centuries later. He led Durin's Folk back to recolonise Khazad-dûm during the Fourth Age, where they remained 'until the world grew old and the Dwarves failed and the days of Durin's race were ended.' The status of this story is unclear, though: it never appeared in The Lord of the Rings, and may have been rejected by Tolkien after its composition.