Dwarves migrate to MoriaDwarves migrate to Moria - Just as the Elves and the Edain fled Beleriand at the end of the First Age, so too did the Naugrim (Dwarves).  Where did they go?  To the ancient city founded by their father, Durin the Deathless.

 

"After the end of the First Age the power and wealth of Khazad-dȗm was much increased; for it was enriched by many people and much lore and craft when the ancient cities of Nogrod and Belegost in the Blue Mountains were ruined by the breaking of Thangorodrim." ~The Return of the King (Appendix A: Durin's Folk)

 

"There were and always remained some Dwarves on the eastern side of Ered Lindon where the very ancient mansions of Nogrod and Belegost had been - not far from Nenuial; but they had transferred most of their strength to Khazad-dȗm...The Dwarves of Belegost were filled with dismay at the calamity and fear for its outcome, and this hastened their departure eastward to Khazad-dȗm." ~Unfinished Tales (History of Galadriel and Celeborn)