I am not familiar with Chinese immigration to Thailand so this answer will focus on Chinese migration to Singapore and Peninsula Malaysia.

There are records of Chinese settling in Southeast Asia from the 9th century onwards, and in the 2000s there was an influx of Chinese from China into Singapore. However, the vast majority of ethnic Chinese in Singapore and Peninsula Malaysia are the descendants of one particular massive wave of Chinese migration. This began in the mid 1800s as the British exerted more and more control over Singapore and Peninsula Malaya (then jointly known as Malaya), and ended in the 1950s with the founding of the People’s Republic of China and the Malayan Emergency. Over this roughly 100-year period, there was strong and broad-based demand for manpower in Malaya, and strong incentives for the Chinese to migrate. Both trends were further encouraged by serendipitous world events.

Many of these migrants still saw China as their home and had strong links with their hometowns and relatives. The period ended with Malaya making a sharp break from China, which played a strong part in the way the Chinese in Singapore and Malaysia think of themselves today, which is to say they think of themselves as Singaporeans and Malaysians, and not at all as China Chinese.

Prologue: British Colonisation of Singapore

An initial wave of Chinese migration began with the British establishing a trading port at the mouth of the Singapore River in 1819, seeking to control the high-traffic Straits of Malacca.

Southeast Asia has historically been in a unique position in the maritime trade route between India and China.

For 6 months of the year, the monsoon winds blow from India, through Southeast Asia and then to China, allowing Indian traders to bring their goods to the region (specifically, the Straits of Malacca) and buy Southeast Asian goods. Thereafter the winds switch direction, allowing the Indian traders to return home and the Chinese traders to head to Southeast Asia to sell their goods, to buy Southeast Asian goods, and to peruse the goods that the locals had bought from the Indian traders to sell on.

Singapore is at the southern mouth of the Straits of Malacca, which meant that it was the closest port Chinese traders could call at if they wanted to directly access the India-China maritime trade route.

It also has a naturally deep harbour, which allowed large ships to get extremely close to shore for easy loading and unloading of goods.

Keen to break the hold of the Dutch on trade in the region, the British did everything they could to encourage traders to call at Singapore, most importantly by making it a free port - there were to be no taxes levied on trade, except for specific products such as opium.

Almost immediately, Singapore attracted enormous numbers of traders, including Chinese traders. In the days before steamships were common, the trip to Southeast Asia could only be made once a year, and time in the region was limited before the winds changed. The choice of ports of call mattered immensely, and the above mentioned factors made Singapore a very good bet.

A typical Chinese junk could carry a load of up to 300 tons and around 250 passengers. With those numbers, demand for support was high. Small boats were required to transport goods right up to shore, for example. And then coolies were required to transport those goods from the shore to warehouses. The sailors needed food, drink and entertainment.

At this time, in the 1820s, there were more Chinese (roughly 6,000) than Malays (around 4,500) and Bugis (about 1,200). They were composed of 2 main groups.

The first were Chinese who had migrated to Singapore to seek their fortune, making a living by supporting traders as coolies, boatmen and other trades. These generally came from Guangdong and Fujian, and as there was a ban on migration by the Qing government at this time, these Chinese came mainly through Portuguese-controlled Macau.

The second were the descendants of Chinese who had settled in Southeast Asia between the 10th and 17th centuries. These Chinese, mainly male, had married female Javanese, Batak and Balinese slaves. Over time, their descendants formed a distinct cultural group - the Peranakan Chinese - clustered mainly in Penang, Malacca and Medan.

The Peranakans in Malaya were well positioned to take advantage of trade in the area. Amongst themselves, they spoke a creole language, a form of Malay that had numerous Hokkien loanwords. This, along with their frequent contact with both the local Malay population and Chinese traders, allowed them to pick up both Hokkien and Malay and act as middlemen between the two.

After the British took control of Malacca from the Dutch in 1824, Peranakan Chinese migrated between the 3 Straits Settlements of Penang, Malacca and Singapore. The Peranakans of Penang were at a particular advantage - having been under British colonial rule since 1800, they had learned English as well, giving them access to Malay, Chinese and British trade. Some of them became very wealthy indeed.