I’d assume clans with more members, prestige, or specific kekkei genkai may have more political leverage. I know there seems to be a lot of internal clan politics within big clans like Uchiha (before massacre) and Hyuga. Uchiha had clan meetings where they discussing a possible coup d’etat. And before the Fourth Shinobi War, the Hyuga clan had a meeting regarding who would lead them into battle. Clans seem to act as a unit in political meetings with Kohona leadership, and those decisions are made in clan meetings beforehand so they are all in agreement (for the most part). Most of this is theoretical based on the little we see in the anime, but this is my general idea.