In The Hobbit chapter 15, while Thorin is talking to Roäc, he says
Also I would beg of you, if any of you are still young and strong of wing, that you would send messengers to our kin in the mountains of the North, both west from here and east, and tell them of our plight.
We know that the dwarves of Dáin Ironfoot live in the east, but who of Thorin's kin live in the northwest?
Appendix A tells us that the Grey Mountains were colonized by Thorin I, great-grandfather of Thorin Oakenshield, following the loss of Moria (emphasis mine):
[T]he glory of Moria passed, and its people were destroyed or fled far away.
Most of these that escaped made their way into the North, and Thráin I, Náin's son, came to Erebor, the Lonely Mountain, near the eastern eaves of Mirkwood, and there he began new works, and became King under the Mountain. In Erebor he found the great jewel, the Arkenstone, Heart of the Mountain. But Thorin I his son removed and went into the far North to the Grey Mountains, where most of Durin's folk were now gathering; for those mountains were rich and little explored.
Return of the King Appendix A "Annals of the Kings and Rulers" III "Durin's Folk"
However, we're also told that these settlements were substantially abandoned after being attacked by dragons:
But there were dragons in the wastes beyond; and after many years they became strong again and multiplied, and they made war on the Dwarves, and plundered their works. At last Dáin I, together with Frór his second son, was slain at the door of his hall by a great cold-drake.
Not long after most of Durin's Folk abandoned the Grey Mountains.
Return of the King Appendix A "Annals of the Kings and Rulers" III "Durin's Folk"
All of these events take place long before the events of The Hobbit; Appendix B tells us that Dáin's death occurs in TA 2589, while the Quest for Erebor occurs in TA 2941.
Since the text above only states that the Grey Mountains were abandoned by "most" of Durin's folk, we're presumably meant to understand that some holdouts remained, even 400 years after the fact.
IMHO, when Thorin said:
Also I would beg of you, if any of you are still young and strong of wing, that you would send messengers to our kin in the mountains of the North, both west from here and east, and tell them of our plight.
Thorin did not mean the mountains northward in latitude from Erebor. Instead Thorin meant the mountains in the northern part of the world, both west and east.
As far as I know, Thorin's only kin in the eastern mountains of the north would have been the Dwarves of Dain in the Iron Hills, and his only kin in the western mountains of the north would have been the Dwarves of Durin's line living in the blue Mountains west of the Shire. That is where Thorin and his companions had come from when they passed through the Shire and picked up Bilbo.
There may have been some other groups of Durin's race in other isolated mountains or mountain ranges in the North.
I don't know how Thorin expected Dwarves from the Blue Mountains to get there in time, considering how long it took Thorin and Company to make the trip. Perhaps the Dwarves remaining in the Blue mountains had refused to try to kill Smaug or steal treasure from him, but said that if Thorin somehow managed to get rid of Smaug they would come in a vast horde with magical speed to help defend Erebor against any ordinary foes like Orcs or Elves or Men.
