The following quote is from Peoples of Middle Earth. It was probably omitted from Appendix A.
Quote:
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And the line of Dain prospered, and the wealth and renown of the kingship was renewed, until there arose again for the last time an heir of that House that bore the name of Durin, and he returned to Moria; and there was light again in deep places, and the ringing of hammers and the harping of harps, until the world grew old and the Dwarves failed and the days of Durin's race were ended.
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So, provided this quote is authorative, the dwarves are led back to Moria by Durin VII (the last incarnation of Durin), and Moria is restored to some of its former splendor, for a while. I expect this happened some time during the fourth age.
-Voronwë