The Unified Clan was the result of the four Original Clans travelling side by side for two decades.
Ironically, this Clan would not have come into existence if not for Tigerstar's, a ShadowClan leader, death.
Within the Unified Clan were four leaders. Each leader had equal control over the entirety of the Clans. Regardless, it was sometimes protested by other leaders that they didn't have as much control as another (the 'another' typically being from either ShadowClan or ThunderClan).
When the Clans first began travelling together it wasn't as a single Clan. Each Clan was still quite separate in leader, mannerism, structure, and blood. Scuffles would frequently break out.