What was the dynasty when Chinese traditional culture was introduced to some neighboring countries and regions?

Korea was a vassal state during the Shang Dynasty, and Japan was enthroned during the Han Dynasty at the latest. Qin Shi Huang occupied Vietnam, and the Han Dynasty set up the county of Jiaotong. The Han Dynasty's Western Territory Protectorate included those countries in Central Asia, such as Kazakhstan, etc. The Han Dynasty's Western Territory Protectorate included those countries in Central Asia, such as Kazakhstan, etc. During the Three Kingdoms, Shu Han's territory included part of today's Burma, and Vietnam was occupied by Wu. Mongolia and North Asia were already under the jurisdiction of the Northern Wei Dynasty.