How did the yamato clan gain control of much of japan?

The took land by force. They also took control of land through marriage. They used iron and irrigation to build their economy.

The Yamato clan dominated an area on
Honshu that became the central area of Japanese government. The clan also established Japan's first and only dynasty. by sending nobles, students, monks, traders and officials to the Chinese court.

The Yamato period (better known as the Kofun period) is the period of Japanese history when the Japanese Imperial court ruled from modern day Nara prefecture, then known as Yamato province. While conventionally assigned to the period circa 250 - 710 CE, the actual start of Yamato rule is disputed.

The Yamato Clan was known for their bravery in battle and their superior fighting techniques. As a result, all other clans within Japan became subject to them. The chief of the Yamato Clan became the first emperor of Japan.