Short answers, pulled from the story.
Buddhist monks traveled with merchant caravans along the Silk Road, beginning in the 1st century CE. Extensive contact increased in the 2nd century CE when the Kushan Empire under king Kanishka expanded into the Tarim Basin, bringing Central Asian missionaries to the Chinese capitals of Loyang and Nanjing. The first missionaries and translators were Parthian, Kushan, Sogdian, or Kuchean.
An Shigao, a Parthian prince-turned-monk, made the first known translations of Nikaya Buddhist texts into Chinese from 148 to 170 CE. The Kushan monk Lokakshema followed, active from around 164 to 186 CE, and became the first to translate Mahayana scriptures into Chinese. Thirty-seven early translators of Buddhist texts are known by name.
The Battle of Talas in 751 CE between the Abbasid Caliphate and the Tang dynasty accelerated a decline that had begun with the Muslim conquest of Transoxiana. Political changes, shifts in trade routes, and new alliances reduced Buddhist activity in Central Asia. In regions such as Iran and Afghanistan, rulers transferred patronage from Buddhist monasteries to Islamic institutions, and without state support, Buddhist sites were gradually abandoned.
Xuanzang was a Chinese Buddhist monk who traveled from 629 to 644 CE, spending 17 years crossing Central Asia and India in search of Buddhist scriptures. He documented his journey in Records of the Western Regions. The texts he brought back launched a "new translation period" that reshaped Chinese Buddhism.
Buddhism arrived in Korea in the 4th century during the Three Kingdoms period. The monk Sundo brought it to Goguryeo in 372 CE, Malananta brought it to Baekje in 384 CE, and Ado brought it to Silla. Japan then received Buddhism from China and Korea. Five monks from Gandhara are recorded as traveling in 485 CE to a country described as "the extreme east beyond the sea," believed to be Japan.
During the 5th and 6th centuries CE, merchants funded Buddhist monasteries along the Silk Roads because those institutions served commercial functions, including lodging, storage, and organized marketplaces. In return, the monasteries provided rest stops for traveling merchants. As merchants moved through new communities, they carried Buddhist ideas with them, and the communities that grew around monasteries became centers of literacy and trade.