Pivotal event for the traditions: King Ashoka and his influence
Ashoka, sovereign of the Mauryan Empire was a key figure in the transition of Buddhism from a local religion to a force that transformed Asia. Ruler of much of the Indian subcontinent from 260 to 232 B.C.E., he encouraged the spread of Buddhism by having edicts containing basic Buddhist teachings carved on Pillars (see Source 2:) throughout his empire and beyond.
Ashoka gave also financial support to monks, built monasteries, and made pilgrimages to important sites in the life of the Buddha, such as his birthplace or the place of his Enlightenment , Following his lead, pilgrimages to these sacred places became a favored practice among Buddhist devotees, and large monasteries grew up on these sites.
As Buddhism spread throughout Asia, rulers of other countries took Ashoka as a model of exemplary Buddhist ruler. In powerful kingdoms within the countries  now called Sri Lanka, Burma (Myanmar), Thailand, Cambodia, Laos, Korea, Japan, and Tibet, Buddhism was established and grew with the benefit of official support similar to the support that Ashoka had given. Buddhism was attractive to rulers because Buddhism was popular with the people, and because support of Buddhism could help legitimate their rule.
Some rulers thought of themselves as as Chakravartin, or "wheel-turning kings". The term in pre-Buddhist India first referred to a universal ruler who would bring morality and peace to the entire world. Within Buddhism, the term originally to the Buddha as universal monarch whose teachings would spread in every corner of the world. The term later was applied to Ashoka, who had spread the
Dharma but continued to rule as a king.

Pivotal event for the traditions: Diffusion in Asia
Buddhism flourished during a period of significant cultural change in Asia. During the first millennium  CE small kingdoms were being absorbed by powerful rulers. Some of these rulers controlled vast territories, as did Ashoka. Qin Shi Huangdi became China's "first emperor" in 221 BCE. Along with this process of political consolidation came the growth of urban centers and trade-based economies.
While many Asian rulers favored Buddhism, it was not spread by armed men conquering territory or demanding conversion. In fact Buddhism provided these new urban centers with cultural support in the form of new social identities, new languages, and new institutions. Buddhist scholasticism was a driving force behind the spread and growth of Buddhism in Asia. As the religion moved into new territories where there was a desire to read and study Buddhist texts, monks became involved in massive translation projects. Chinese monks made a long journey to the large university-monasteries India to obtain texts to be translated. Monks from South Asia also traveled along the Silk Route to China to teach and aid in translation of texts. Statues and other forms of Buddhist art were carried through countries as well becoming visual means of introducing Buddhism.
In many areas where Buddhism was introduced it was a civilizing force, bringing with it elements of the sophisticated cultures of the South Asian subcontinent, and later, of China. Buddhism had an aura of power, partly because it had the support of powerful political forces, and partly because Buddhist monks were recognized for their scholarship. Monks were also thought to have special powers such as the ability to heal sickness, predict the future, and control rain.
Each region to which Buddhism traveled developed its own monasteries,  temples, new schools and lay following. With time Buddhism would develop a unique and different character in different Asian countries, also thanks to its capacity of merging with pre-existent cults (see Buddhist Module I p. 6, intercultural and interdisciplinary information and p. 9 spirit-related practices).
 Southeast Asian Buddhism is believed to most closely resemble early Indian Buddhism, and Southeast Asia is now the center of the Theravada Buddhist tradition. China, where Mahayana Buddhism has flourished, became a center for the transmission of Buddhist religion equal to and eventually surpassing India. Tibet, influenced by both Indian and Chinese
Vajrayana Buddhism, developed a Buddhist theocracy and was ruled by Buddhist monks until the Chinese invasion of the mid-20th century. Chinese Buddhism was transported to Korea, Japan, Tibet, and Vietnam, and each country developed its own particular forms.
By the end of the 12th century C.E., Buddhism had virtually died out in India. A variety of factors are responsible for this, including a resurgence of Hinduism, which incorporated many of the innovations of Buddhism, and Muslim invasions.

Sources' analysis

Source 1:
Buddhism expansion in Asia.
See above "Pivotal event for the traditions: Diffusion".
For a brief description of the characteristics of Theravada, Mahayana and Vajrayana Buddhism, see p. 1 of the  Module I on Buddhism.

Source 2:
The Lion Capital of Ashoka.
The Lion Capital  features four Lions standing back to back. They are mounted on an circular base carrying sculptures in high relief of an elephant, a horse, a bull, and a lion, separated by intervening spoked chariot-wheels. The whole sits upon a bell-shaped lotus.
The lion is frequently used as a symbol of the Buddha, and the spoked chariot-wheels, called Ashoka Wheels, are symbols of the Buddha's teaching.
The language of the inscriptions on various Pillars varied according to where they were located; there are even ones written in Aramaic and Greek. The inscriptions revolve around a few recurring themes: Ashoka's conversion to Buddhism, the description of his efforts to spread Buddhism, his moral and religious precepts, and his social welfare program. The inscriptions also advocated religious tolerance and announced that he had sent people to teach the Dharma across his kingdom and beyond.
A graphic representation of the Lion Capital was adopted as the official Emblem of India in 1950, and the Ashoka Wheel from its base has been placed onto the center of the National Flag of India.
See http://india.gov.in/india-glance/national-symbols

 

Link to other Digital Modules
Module on Christianity I sec. 3 and 4
Module on Islam I sec. 2  and 3
Module on Migration and Diaspora
Buddhism Module I sec. 6 and 9