I'll stand in for Kuru.... I'm using a Kuru-esque avatar, so that'll be permissible, methinks.
Starting in the First Age, each of the seven houses of Dwarves had a sort of "mountain range" they called home. Starting in the west, we have:
The Firebeards and the Broadbeams, or the Dwarves of Nogrod and Belegost, in the Blue Mountains. These are the Dwarves we see in the
Silmarillion. The Dwarves of Nogrod were decimated by the Elves, after their major quarrel with Thingol, and both mansions suffered greatly in the breaking of Beleriand at end of the First Age. The survivors (mostly from Belegost), mainly moved eastwards to Khazad-dûm, and joined the people of Durin. The Dwarves of the Blue Mountains, in later ages, were mostly Dwarves of Durin's race who moved back into the area after troubles in Moria and Erebor.
The Longbeards, or people of Durin, dwelt in Moria in the Misty Mountains. In their early years they occupied all the Misty Mountains, including Mt. Gundabad, as well as eastwards to the Iron Hills, where they would continue to have a presence until the Fourth Age, at least. After the loss of Khazad-dûm in the mid-3rd Age, the Longbeards removed to Erebor, by way of the Grey Mountains. From their ancestral domination of the mountains around the Wilderland, and from this time of wandering, it is clear that any Dwarves in the Grey Mountains would of necessity have been Longbeards (and the intermingled Dwarves of Belegost descent, by this time).
East of the Iron Hills, we know little of the last four houses of Dwarves. Nearer to the Iron Hills dwelt the Ironfists and the Stiffbeards, and furthest away dwelt the Blacklocks and the Stonefoots. We know nothing of their locations save that they were at distances "as great or great than that between the Blue Mountains and Gundabad". These Dwarves seem the most likely to have provided the few Dwarves that served with Sauron in the War of the Last Alliance, and also provided aid to the Longbeards in the days of Thraín II, at the Battle of Azanulbizar. They also are referenced, most likely, in the "The Shadow of the Past":
Quote:
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There were, however, dwarves on the road in unusual numbers. The ancient East-West Road ran through the Shire to its end at the Grey Havens, and dwarves had always used it on their way to their mines in the Blue Mountains. They were the hobbits' chief source of news from distant parts---if they wanted any: as a rule dwarves said little and hobbits asked no more. But now Frodo often met strange dwarves of far countries, seeking refuge in the West. They were troubled, and some spoke in whispers of the Enemy and the Land of Mordor.
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--emphasis mine.
Up to the italicised line, when Tolkien talks of dwarves using the East-West Road, he would mostly be talking of the Longbeards, who had resettled those mines in the Blue Mountains, and who would have been travelling back and forth to Erebor (or in earlier times, Moria). However, the line that I have italicised clearly deals with new, different Dwarves "of far countries", who are clearly not the Longbeards familiar to the Hobbits. Coming from east of the Iron Hills, they fit both the profile of "strange Dwarves" and would certainly have been in lands being once again dominated by Sauron.
So, there you have it: my "short" discursus on the different Dwarves and their homes. The above is all taken (with the exception of the
Fellowship of the Ring quote) from HoME vol. XII
The Peoples of Middle-Earth, from the essay entitled "Of Dwarves and Men". It dates to around the time Tolkien was working on the Appendices, during the publication of the LotR, and is, as I recall, never contradicted in a later statement, and can be taken as pretty much canonical.