This answer of /u/SleepingAran/ answers your question: https://www.reddit.com/r/AskHistorians/comments/4txvom/why_didnt_hinduism_buddhism_taoism_or_any/

So, I'm seeing a lot of misinformation here and would like to clear some things up. To do this, I'm going to run through your question's misunderstandings and then move to the things I've seen that are wrong in other replies before addressing the question directly. I should also state, as I've received some PM's accusing me of bias, that I'm a practicing Therāvada Buddhist as well as a Buddhist academic working primarily within the framework of the Paḷi textual tradition as well as comparisons to the Āgamas.

The error I notice in your question is the presumption that the Aśokan missions found incredible success in the the East and in South-East Asia. While many Southeast Asian countries will attribute their Buddhism to Aśoka, the evidence simply doesn't indicate such an early adoption of Buddhism, or at the very least that it was that widespread. In the East, that is Gandhāra, Buddhism seems to have already been well established among monastics. There is some indication that the Greeks during Chandragupta Maurya's time knew of the sect, at least knowing the distinction between Brahmanas and Śramaṇas and that Śramaṇic religions were well established in Takṣaśilā. Buddhism had a gradual spread into Central Asia, and eventually East Asia from it's inception to the second or third century CE when it became dominant in the Tarim Basin and in parts of China. As for Southeast Asia, it may have found a foothold, but we don't have much evidence for it until the 1st through 4th centuries CE, by which it seems to have become the lead religion of the Mon and Pyu peoples, while elsewhere forms of Hinduism and native religions were most common. The modern ethnic and cultural composition of the mainland is fairly recent due to population movement since the medieval period. The Buddhification of Southeast Asia was a slow and gradual process over the course of more than a thousand years before dominating the region in the early medieval period after royal patronage of Hinduism largely ended and Buddhism was adopted and an orthodoxy established. This was done through sustained contact with the Buddhist heartlands in East India and Śrī Laṅka.

The claims I'm seeing here are basically two things. One, Aśoka had little contact with the Western Greek kingdoms and two that it had followers far away from India and that somehow Christianity or persecution ended it. Let's start with the first one; In his 13th rock edict (or what we call the 13 rock edict, there were probably a lot more than we know about and we really can't establish a timeline on them.) Aśoka speaks thus:

Now it is conquest by Dhamma that Beloved-of-the-Gods considers to be the best conquest. And it (conquest by Dhamma) has been won here, on the borders, even six hundred yojanas away, where the Greek king Antiochos rules, beyond there where the four kings named Ptolemy, Antigonos, Magas and Alexander rule

These are absolutely real Greek kings who rule at his time, Antiochus being Antiochus I Soter of the Seleucid Kingdom in modern day Iran, Iraq, and the Levant; Ptolemy being Ptolemy II of Egypt, Antigonus being Antigonus II of Makedon, Magas being Magas of Cyrene (a briefly independent state in modern day Lybia that I had to look up, and Alexander probably being Alexander the II of Epirus. What we can gain from this, at the very least, is that Alexander was in contact with the Seleucids enough to know of key diplomatic rivals. We also know through Seleucid records that there were fairly substantial diplomatic ties between them and the Mauryans even fairly late into their existence due to a several travel guides, of which only one can be reconstructed. Megasthenes' Indika was followed up by other diplomats writing travelogues and it seems to contain accurate geographical and cultural information recorded during the reign of Chandragupta Maurya. Furthermore, Pliny mentions that Ptolemy had diplomatic relations with Aśoka who wrote his own (now lost) travelogue.

A more credible criticism is that Aśoka's Dhamma envoys were secular or at least not Buddhist. This is something that can be neither proven nor disproven. In another edict he mentions that he has sent his Dhamma Mahāmātās to religious disputes, but in yet another he sends them to oversee all religions and to oversee ethics. In a recent argument, Dr. Nayanjot Lahiri argued that these were Buddhist missionaries, repurposed as diplomats and ethics instructors. While a somewhat interesting argument, it is mostly conjecture based on the highly biased account given in the Mahāvaṃsa. In any case, we can at least say there WERE missions under Aśoka and there was some religious dimension to his Dhamma. However, the farther away from Aśoka's core territory they were sent, the less successful they could have been and the more difficult it would be to convert people, the exception being Śrī Laṅka which absolutely did see missions and successful ones under Aśoka.

Because of this, I highly doubt there was ever a Buddhist presence beyond maybe one or two diplomats or traders in Europe, at least until the eighteenth century. Which more or less handles the second concern I had.

Now why didn't Aśoka see much success outside of his core realm. I think it is relatively simple. People are typically pretty happy with how they are and unless there is some impetus to convert, be it economic, political, or environmental, they won't. While there will always be outliers who find themselves intellectually interested in a religion, they are often far and few between. Sending a missionary from one powerful ruler to another doesn't really change much, nor does preaching to a people who are quite happy with the status quo or are already experiencing religious oppression. It is said that Devānaṃpiya Tissa adopted Buddhism (and the name Devānaṃpiya) after meeting with Aśoka's son, Mahendra, who had become a monk. This is actually supported by some evidence and it wouldn't be all that out there for a man, struck by the conquests, riches, and power of another king, who still appeared to be somewhat humble and paternal, to convert upon being asked and given gifts. It would hardly be the only time in history it had happened. And once the King converts and begins to patronize a religion, well there's the incentive for the rest of the populace.

Ultimately, Aśoka's missions may have had an impact among the Pyu or Mon or whoever inhabited the realms to the direct east of his empire, but since they left scare records and remains, we can't really know. Within India, he may have been Buddhism's greatest patron, but outside the process was rather gradual.

Sources

Aśoka in Ancient India Nayanjot Lahiri, 2015.

The Edicts of King Aśoka S. Dhammika, 1993.

Mahāvaṃsa William Geiger, 1912.

A History of Myanmar Since Ancient Times Michael Aung-Thwin, 2012

Indian Buddhism AK Warder, 2004

A Concise History of Buddhism Andrew Skilton, 2013.