Short answers, pulled from the story.
Zhang Qian departed the Han capital of Chang'an in the year 138 BCE to secure alliances against northern nomads. His journey across the Eurasian Steppe opened direct contact between China and Central Asia for the first time in recorded history.
An Shigao arrived in Luoyang in 148 CE as a Parthian prince who had become a monk. He translated basic doctrines, meditation techniques, and abhidharma texts into Chinese, establishing the first known wave of Buddhist proselytism in the region.
After the Battle of Talas occurred in 751 CE between the Abbasid Caliphate and the Tang Dynasty, local Tocharian Buddhist culture in the Tarim Basin went into serious decline and eventually vanished by the eighth century. By the end of the eighth century, regular exchange of Buddhist texts stopped entirely due to Muslim conquest of Transoxiana.
Mahayana Buddhism arrived on the Korean peninsula during the third century CE from China via Tibet with Malananta bringing teachings to King Chimnyu of Baekje in 384 CE. Sundo introduced them to Goguryeo in 372 CE while Ado later transmitted Buddhism to Silla in central Korea.
Zhang Qian departed the Han capital of Chang'an in the year 138 BCE on a mission to secure alliances against northern nomads. His journey across the Eurasian Steppe opened direct contact between China and Central Asia for the first time in recorded history.