Firstly, we tend not to use the term 'Nestorian Christianity' and 'Nestorian Church' these days. It's inaccurate and misleading.

It's geographical spread, as your linked map shows, was extensive, though its presence there as never as dominant as Europe.

In the case of China, the Church of the East (CotE) disappeared beginning in the 9th century. By 987 we have an Arab report stating that Christianity was virtually extinct in China. This is matched by the absence of any mention of Christianity in the 11th through 13th centuries.A number of theories have been advanced. Notably, there was a rise in persecution of foreign religions under the T'ang dynasty in the mid 9th century. Legge argued that it was theological syncretism that led to its demise, but based of the documents we have this does not seem to be the case. Moffett suggests that it is simply the failure of the T'ang dynasty itself, which provided religious freedom. With that gone, so too did the protections that enabled Christianity to gain a hold in China. Christianity did return to China under the Mongols, but seems to have disappeared with the collapse of the Yuan dynasty and the rise of the Ming.

The main reason for the decline in the CotE throughout the rest of Asia (excepting India) is the rise of Islam. Although Christianity occupied a protected place as a minority in some times and places, it still existed as a minority. And it exists as a minority for a long time. At times this took particular persecution, but not always. But, for example, al-Mutawakkil (Abbasid caliph) in 850 controlled who the patriarch was, destroyed churches and monasteries, forced Christians to wear yellow clothes and patchs, forbid them to learn Arabic, destroyed graves, and denied them places for worship.

This was not the universal experience though. The CotE also seems to have, around the 9th-10th century, lost its impetus for expansion from internal reasons that remain unclear.

One should also not exclude the impact of the Mongols(!) While Chinggis was known for general toleration of religion, and the CotE extended to the Mongol heartland, the widespread devastation of the Mongol invasions damaged the fabric of the societies they attacked, and this disrupted the sometimes niche position Christians occupied. More long-lasting though, the successor khanates that sprung up after Chinggis through Central Asia tended, one by one, to convert to Islam. Thus, again, ending the religious protections Christians may have enjoyed. (This is true for the Ilkhanate, the Chaghadai, and even the Golden Horde, in the 4th century. Christianity among the Keraits and Uyghurs does not appear to have survived into the 15th century).

India is its on situation, with which I am less well versed. Suffice to say (here) that it did not disappear in the same way the CotE disappeared elsewhere. The Thomistic churches survive until today.

For a populist history book dealing with this, Philip Jenkin's The Lost History of Christianity is pretty good. For a more detailed exploration, I recommend Samual Moffett's A History of Christianity in Asia.