This is how the clans get along:
- The Bogbeards live in dank, substandard caves because the Axehammers drove them out of their ancestral lands. But the Axehammers never would have invaded Bogbeard territory if the latter had honored the Treaty of Blackanvil and supported the Axehammers in Goblin War VII. The Axehammers lost and now goblins live in their halls. Both sides are haunted by grumpy ancestor spirits, who constantly complain their grandkids have abandoned the ancient homelands.
- When the Dwarf Throne sat empty 7 centuries ago the Electrumeyes and Flintdoodles both made plays for the kingship and both ended up ruining each others' chances. Now every time the fortunes of the Electrumeyes clan goes pear-shaped they all remind themselves that they'd be living the easy life if those jerks the Flintdoodles hadn't interfered, and vice versa.
Now, in addition to a little extra texture to dwarf-on-dwarf encounters, you probably also have the makings of one or more epic quests. Any PC that can unscrew centuries of dwarven grudgery deserves mountains of accolades.
The same basic system works for halflings, but make sure the grudges are over the smallest, pettiest stakes possible. E.g. the Bottomrumps and the Dingleberries argue every year over who is in charge of the Christmas pageant.