In the film-canon, the other clans felt that the risks of helping Thorin outweighed the potential benefits--possibly because King Thror had become so stingy with his treasure--though they would have presumably united under Thorin Oakenshield if he could possess the Arkenstone. The films make the Arkenstone into a symbol of the King under the Mountain that seems to grant him the authority of a High King over all of the Houses of the Dwarves.
Erebor was home to the Longbeards, so the quest was their own affair. Bifur, Bofur, and Bombur were not from Durin's line, although their families came from Moria. So it is possible they may come from other clans who went to Moria after their homes in the Ered Luin were destroyed.
In my head-canon Bifur, Bofur and Bombur are Broadbeam Dwarves or are at least descended from the Dwarves of Belegost, though they could (as you say) be Longbeards who are not part of Thorin's family line.
Bombur certainly put the broad in Broadbeam!