Blue Mountain Dwarves
 

In the Ered Luin there were the twin dwarf cities of Nogrod and Belegost. Did these cities get destroyed after Beleriand was destroyed? If they were then where do the dwarves that live in the Blue Mountains during Frodo's time live?

Yes, they were. Then the survivors migrated to Khazad-Dum.... or to other Dwarvish cities, since Khazad-Dum was the mansion of the Longbeards, of Durin's Folk. The Dwarves that lived in the Ered Luin during Frodo and Bilbo's time were Durin's Folk, or at least their survivors.

That's from the first movie, stated by Saruman to Gandalf; and it's pretty accurate. THe Dwarves were attacked by Durin's Bane, a Balrog of Morgoth, and the Orcs that Sauron sent. So these Longbeards, led by Thorin's grandfather Thror, founded the Kingdom under the Mountain, and was so rich that it was soon assailed by the dragon Smaug. They escaped to the Blue Mountains and lived a poor life as coal miners...

;)

I think I read this in the LoTR appendices: most, maybe all, of the Dwarves who lived in Ered Luin at the end of the Third Age lived in the range to the south of the Gulf of Lhun. Everything west of the River Lhun was Elvish country under Cirdan's control and may account for why few, if any, Dwarves lived in Ered Luin's northern range.