While I can't exactly fully answer the question with regards to Persia, perhaps u/icnielsen can, I can comment on China (including some Central Asian tributaries) and India.
First lets talk about how the Church of the East became established in these (very large) regions. After all, it would be a fools errand to talk about the end of a story without knowing the begining.
India in the third century CE was an exciting and important place. After a chaotic period of clashes with invading forces from Central Asia, the Kushans had risen to power and established a large empire covering North India and Central Asia. Not only was their territory expansive, but it was very, very rich. Along with the Parthians, the Kushans controlled the Land Silk Road and had a foothold in the Maritime. When combined with their somewhat unusual location and the already cosmopolitan population of Central Asia, this made them prime area for the spread of a new religion. Thus the Church of the East entered India, and to their surprise, or perhaps not, they found small communities of Christians in the south.
According to the Apocryphal Acts of Thomas, compiled in the third century, Thomas ventured to India and converted an Indo-Parthian King called "Gundaphar." Eventually he ventured south into India proper where he was eventually executed. Of course, this is all hagiography but perhaps there is some truth to it. An Indo-Parthian King called Gondophares ruled at the right time, though his coins bear the standard Indian and Greek dieties worshiped in the area, but this does not mean he didn't accept Christianity in some form or was unsympathetic to it, syncretism was abound in the region. Furthermore the Malayalam Thomma Parvam and other Keralan traditions trace a line to to St. Thomas. While these legends are late, it doesn't mean they are worng. I'm honestly unsure of the prescence of Christians in Kerala before the 3rd century, I go back and forth, but nevertheless by the time of the early Eastern Missions, they were there.
When the Church of the East split from Rome, they entered a period of renewed missionary activity and many moved into South India, mainly Kerala, which became the center of Christian activity until the colonial era. Furthermore, there was already a small population of Jews in Kerala and indeed some liturgy of the St. Thomas Christians bears a similar structure to Jewish liturgy unseen in other Christian traditions, suggesting close contact and intermingling. By the middle of the 4th century, there was an official Bishop of India.
Unfortunately for the Syrian church and the Saint Thomas Christians, the chaos in Persia and North India in the 9th-15th centuries made contact between the Syrian Church and India unstable at best. Here's the thing though, Christianity never disappeared. Rather, it remained a powerful local sect and Christians in Kerala were at least powerful enough to be acknowledged in local inscriptions. When the Portuguese sailed to India they were overjoyed to find a thriving Christian commu ity there and communion with the Pope and the remnant Syriac Church was established.
The Portuguese, however, were less pleased with the character of local worship, deeming "Nestorian" and pushing back against it. Whether the Church of the East was truely Nestorian is a matter for Christian scholars, but the relationship is complicated and coloured by Latin heresiology. In any case, this split the community between a number of factions, though the largest was (and remains) the East Syrian faction of the Syro-Malabar, who use the Eastern rite while in communion with Rome. The Faction remaining loyal to the Church of the East is much smaller, but still extant.
In China, missionaries first are recorded as entering in the Sixth Century via the Tang Western Protectorate. Christianity in China was different and deeply transformed by the encounter, Sinicized, Confucianized, even Buddhicized. The Christian manuscripts from Dunhuang suggest a degree of syncretism used to bridge the gap between Chinese and Christian thought. The earliest text, the Sutra of Hearing the Messiah (序聽迷時所經), dates to 635 and shows perhaps the most Chinese influence, referring to Buddhas and Devas. Later discourses, such as those found on the Tomoika manuscript, are more identifiably Christian and far less syncretic in character, possibly due to being aimed at an audience already familiar with Christianity.
Unfortunately, the timing for Christianity to enter China was poor. By the late seventh century it became very difficult to contact the main church in Syria. However, Christianity continued in China and seemex to grow. In 781, the Xi'an steele was erected in the imperial capital. Unfortunately for a great many in China, this religious tolerence was not to last. The Tang emperor Wuzong (武宗) came to power in 840 and fell under the influence of Zhao Guizhen (趙歸真) a militant Daoist monk, when combined with his paranoia and threats from the Uyghurs he and his court issued anti-religious edicts. Early on in 842, these just targeted monasteries and forcibly laicized Buddhist monks, but in 845 these turned violent and the Emperor ordered all "foriegn" religious temples destroyed and looted. This persecution more or less destroyed public worship of Christianity and Manichaeism in China proper, forcing it into the outlying regions of Central Asia, which remained cosmopolitan under the Qocho kingdom and Mongolian Khans.
Christians found a safe have. In Qocho, where the Manichaean Uyghurs settled after the Uyghur Khaganate collapsed. While Ruled largely by Buddhists, it kept the Central Asia tradition of religious tolerance and pluralism that the region was known for. Similarly, Christians found a home among the Mongols and Khitans who came into conflict with the Tang. There is some evidence of there being a small population under the Liao of North China.
Speaking of the Mongols, there are an important part of the story. Chinggis Khan's empire united Central Asia, Persia, and Syria under a single banner for the first time in history and allowed the Church of the East and reestablished Christianity in China under the Church of the East. The Il-Khanate was quite favorable to Christians, even following the conversion to Islam that led to the Buddhist purges and things were even more favorable in China under the Yuan. However, this too was not to last.
The collapse of the various successors of the Great Mongol empire led to a rise of xenophobia across the board. China was no exception. The Ming who ruled after the Mongol Yuan were rather intolerant of things they viewed as forien, including Syriac Christianity and Manichaeism. This led to the final decline of both religions in China and the final decline of the later in the world. This, unlike the Tang persecution, was not violent, but rather slow and gradual with Christianity absorbed into the fold of Chinese syncretic salvationist religion and the structures destroyed bh time or repurposed. Christianity would again trouble China in the late 19th century, but the Church of East's formal history in China ends in a slow decline, forgotten by the annals of history.