The Dwarven population probably eventually died out for two main reasons. First off, there were few females, and not many males were willing to be married. That itself reduces the number of offspring produced drastically. Secondly, the Dwarves kept to themselves, of course with a few notable exceptions, ergo their beliefs/customs/ideas weren't spread out over an area. Since no one really knew much about them, they quickly began to be forgotten.
By the end of the Third Age, the only Dwarf kingdom that existed in Eriador was the Lonely Mountain, ruled by Dain II Ironfoot and later, his son, Thorin III Stonehelm. There was a smaller settlement at Aglarond, "ruled" by Gimli, son of Gloin, but that wouldn't really count as a kingdom, per se. It would be more of a settlement of Dwarves working together. I don't think Moria was re-populated because of the shadow that it had long lain under. It would take a very long time for people to forget that shadow. Even though there are only two main "settlements" of Dwarves in Eriador, there could still be a lot more further East into Middle-earth. They just don't come into the story.
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But Melkor also was there, and he came to the house of Fëanor, and there he slew Finwë King of the Noldor before his doors, and spilled the first blood in the Blessed Realm; for Finwë alone had not fled from the horror of the Dark.
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