I can't speak to the specific communities, that is, I cannot tell you any sort of percentage breakdown by chronology. However, it is important to remember that Chinese had been a major presence in the Indian Ocean since at least the Song Dynasty--the stereotype of China being "closed off" to foreigners is more or less a complete fiction invented by the European colonialists. A good illustration of this is found in the Portuguese port archives for Malacca, which recorded the origins of the various ships who docked there and passed through. The great majority, of course, were local, but of those that were not local the only origin point that can't be slotted into "other" was China, which accounted for something like 25% of all traffic (I don't have the source in front of me). So Chinese ships under Chinese merchants were a major part of the picture of modern commerce in the Indian Ocean.

How does this translate to actual movement of people? There are a variety of methods. The simplest is diplomatic exchange: it was not uncommon for Chinese princesses to be married to nearby rulers. This would involve a rather significant number of people, as servants, handmaidens, lesser relatives, and a varied crowd of hanger-ons would accompany them. More significant would be the trade diaspora (although I sense this term is being replaced by ideas of "networks and circulation societies"), which would be in a feedback loop with those in the diplomatic exchange. Even today someone wanting to conduct long distance trade will need a "man on the spot", and this was only more true before modern communication methods. This may start as only a few families, but it would blossom as hanger-ons, craftsmen, clerks, minor associates and the various other people required to run a trade post gather to the settlement. This may then act as a magnate that attracts further migration from people who decide there is opportunity involved in the trade. perhaps your cousin is doing well in Majapahit and so you decide you can do better there than in Fuzhou. This is an age old process, visible the world over and still ongoing to this day.

However, I believe the majority of overseas Chinese are descended from the waves of migration that have been set off by the various instabilities of the last few centuries, particularly the bloodshed and violence of the nineteenth century. Using the well established trade connections discussed above, refugees hoped to flee the violence that engulfed southern China by settling in SE Asia. Their relationship with the Chinese government and the non-Chinese locals both, particularly over the last century, has been rather complex, as you no doubt know.

I am certain there is a good book covering this, but I am getting my information from Jacques Gernet's History of Chinese Civilization, Barnes and Parker Ships and the development of Maritime Technology in the Indian Ocean and Chadhury and Morineau Merchants, Companies and Trade.

I'm neither a historian nor a specialist on ethnic Chinese diaspora, but i'm Southeast Asian Chinese myself (actually, i prefer to say Chinese Southeast Asian, but this sounds slightly awkward to my ears), and currently working on a Masters thesis project that happens to concern the Chinese in Malaysia and Singapore.

The questions you're asking are very broad, and i'm sure others can do a better job than me of answering them, but here are some quick and easy answers i can give:

The Chinese arrived in SEAsia as a result of junk trade (by which i'm sure you know i mean trade on junks, not trade of junk), originating primarily in the provinces of Fujien and Guangdong, and encouraged by European colonial rule. By the 16th century they had established permanent settlements in various parts of SEAsia. The largest numbers settled in Thailand, Malaya, and Java, although it should be said that they are represented in most other areas in the region as well, including the Philippines, Borneo, Vietnam, and of course, Singapore, where they form a large majority of the population today.

It's hard to say how many there were/are, because it's hard to define "Chinese". Prior to the 20th century there was a tendency amongst them to assimilate with the locals. If we take self-identification as a criteria, by around 1959 there were about 10 million of them in the region. It should also be said that by the 1950s mass migration from China had basically ended, and most growth following arose from local births.

I'm taking all of this from Skinner's 1959 article, "Overseas Chinese in Southeast Asia", which i just happen to be reading now, but there are numerous other sources which i don't have handy at the moment, unfortunately.

EDIT: Spelling

EDIT 2: It just occurred to me to also point out that the question of how many ethnic Chinese there are in Southeast Asia today is complicated by more recent immigration from China. In Singapore, this has actually been actively encouraged by the government in recent years, and their numbers have gotten quite substantial. Many have gained permanent residency status, or even citizenship. Should these be considered "ethnic Chinese in Southeast Asia", then? Culturally, they can be quite different, and Chinese Singaporeans whose forebears arrived earlier in the 20th century generally don't identify with them. I don't have any references for this, unfortunately, but you can check out this article in the New York Times, which is quite interesting in any case.

I'm not familiar with the intricacies of Chinese history, but I'll try to answer the question:

*Where in South East Asia did they emigrate?

Chinese can be found all over South East Asia. There are communities in Vietnam, Thailand, Burma, Indonesia, and Malaysia.

*How did they get there? Why did they go? How many went and how have their numbers changed over time?

It really depends on the time and place. Beginning in the Ming Dynasty (1368–1644), for example, China established a tributary system in which Korea, Vietnam, and Burma acknowledged China's cultural supremacy. In return, the the Ming emperors allowed these states to trade in Chinese lands and recognized the legitimacy of their rulers.

Besides trade, kingdoms such as Vietnam and Burma borrowed heavily from Chinese culture. In Vietnam, for example, the ruling dynasty imported the Chinese script and established a government modeled on neo-confucian principles so popular during the Ming period. Unsurprisingly, these rulers also encouraged Chinese traders to settled in their lands for economic and cultural reasons. Since Chinese merchants and sailors were familiar with Southeast Asia trade routes I believe they would have arrived to Vietnam by sea.

For centuries, Chinese merchants also settled the Southeast Asian islands to pursue trade opportunities. I remember reading that these communities became quite assimilated in the Muslim kingdoms of the region and served in prominent positions in their courts.

I also wouldn't be surprised if Chinese settled in Southeast Asia for political reasons. During the early Qing Dynasty (late 17th century), for example, many Ming loyalists (nobles) and other rebels fled by land into Burma, Vietnam and Korea rather than acknowledge the new Manchu emperor.

The 19th century probably saw the biggest wave of Chinese immigration to Southeast Asia, as well as to USA (gold rush) and Peru (guano mines). A majority of these immigrants were Cantonese. This was because Southern China suffered from widespread famine resulting from the Taiping Rebellion in the 1850s.

The several million Chinese that settled in Indonesia, Malaysia, Vietnam and Singapore would come to play a large part in those countries' mercantile communities.

The King of Siam (Thailand), also imported several hundred thousand Chinese laborers to help modernize the country.

Sources: (admittedly, I'm the first five books I'm listing are from the bibliography of Benedict Anderson's book on nationalism, Imagined Communities. The Chinese diaspora plays a minor role in his argument)

Development of Labor Institutions in Thailand, p. 38

The Chinese in Philippine Life, 1850-1898

Chinese Society in Thailand

Battye, Noel A. 'The Military, Government and Society in Siam, 1868-1910.

Thongchai Winichakul. 'Siam Mapped: A History of the Geo-Body of Siam.

Jonathan Spence, The Search for Modern China

I can tell you about Chinese in Indonesia.

The first time a Chinese ever came to Indonesia was Faxian, a Buddhist monk who stopped by Java island for five months along his way to India. He arrived at Tarumanagara Kingdom, at the time the only great kingdom in West Java.

The second time was also a Buddhist monk, whom we call I-Ching (pinyin: Yijing, hanzi: 义净). Traveling by boat from Guangzhou, he arrived at Sriwijaya (present-day Palembang, Sumatra island) and proceeded to live there for six months. At 687, he returned to Sriwijaya and lived for two years before returning to Tang China (thanks to him we Indonesian got a great insights of their daily lives).

Because Sriwijaya was located in a very strategic position for trading, which was Malacca Strait, many foreign traders stopped by, but until long after Yijing came, Sriwijaya only saw Persian traders.

Chinese traders began coming around ninth century to trade porcelain, silk, tea leaves, and pottery utensils to be traded by spices and bird's nest. But at the time, only men came due to the danger of the journey. There are evidence found revealing few of these men came to stay in Sriwijaya and assimilited with local culture, marrying local women etc.

At 1293, Kublai Khan attacked Java. His troops, led by Ike Mese, arrived in java to avenge his messenger, Meng Chi, who was disgraced by Kertanegara, the king of Singosari at the time. Before they even had the chance to attack, Ike Mese met Raden Wijaya, who hoodwinked Ike Mese by pretending he wished to help Ike Mese, but in reality only wanted to use his troops to regain his throne. Raden Wijaya would become the first king of Majapahit whose territory would include all present-day Indonesia, Sungapore, Pahang, and Papua New Guinea. Believing Raden Wijaya, Ike Mese didn't prepare for a surprise attack commited by his 'ally', causing him to retreat with shame. In this war, thousands of Mongol soldiers were captured. Raden Wijaya let the soldiers who accepted his rules to stay in Majapahit, where they married local women and assimilated with local culture.

Admiral Zheng He came at 1405, bringing Islam as well as Chinese men who also stayed and assimilated, but until that time, no Chinese woman ever came to Indonesia. Several 'kampung Tionghoa' (Chinese communities) that existed up until eighteenth century consisted of the descendants of Chinese men and local women who married amongst themselves, creating what we playfully call hitachi (black Chinese), due to their darker skin than common Chinese. The only Chinese women ever came were royalties, princesses who were married to Javanese royalties at the time, but considering only royalty could marry royalty, it is highly unlikely these royal Chinese descendants still retained their Chinese culture. The kingdom of Demak, first Islamic kingdom in Java, was built by Raden Patah, whose mother was a Chinese royal.

Just a tidbit, we have a temple for Admiral Zheng He in Semarang (my city!) whom we call Dampo Awang. Even though Zheng He himself was muslim, many Buddhist come to this temple to pray. We also keep an anchor from his ship, whish locals believe can give them luck and blessing from the admiral himself.

VOC also has a finger in the pie. Due to the lack of workers, they imported Chinese workers especially for mining in Bangka Belitung area, who then married local women as well. These hitachi (black Chinese) descendants are now commonly known as cina benteng (I don't really like this term because it sounds derogatory...)

(Sad to say that most of pure second generation Chinese who came around 1930s strongly opposed against their descendants marrying these cina benteng, because they are "impure" in their eyes, including my father who thinks of them as 'Chinese traitor')

In 1740, though, VOC massacred more than 7,500 Chinese who rebelled due to their dictatorial rules, causing the indigenous Indonesians and Chinese traders alike living in poverty, you can read my previous comment about VOC's cruelty in East Indies here. This incident is known as 'Red Angke' since the blood of the beheaded Chinese people (men, women, old and young, all were massacred) reddened the Angke River in Batavia.

Around early nineteenth century, Chinese women started journeying to Indonesia along with Chinese men, or to be exact ran away from their country due to Boxer and Taiping rebellion, and also the Opium War. These people started the 中华会 organization, who also built school for Chinese children.

When Japan attacked China and established the Manchukuo, a deluge of Chinese immigrants came to Indonesia (including my grandparents whose village was ravaged by the Japanese). These Chinese are now known as 'first generation' (my generation is the third generation), despite people already coming even before them, because this was the first time in Indonesia's history Chinese immigrants came in a very large number, even outnumbering the cina benteng population, bringing their family, wives, sons, and daughters. The estimated number is around 100,000 people to Batavia only (not mentioning Medan, whose Chinese now speaks Fujian dialect instead of Indonesian, or Singkawang in Borneo, whose Chinese speaks Hakka).

Source:

• Tionghoa Dalam Pusaran Politik

• Hoakiau di Indonesia

• Peranakan Tionghoa di Jawa

I replied to a similar question a while back; you might still find it useful, though I focused on the Chinese diaspora in the Malay Peninsula.