In discussing the boom in trans-Pacific Sino-American migration that took place from the end of the 1840s to the middle of the 1870s, it is worth noting that the forces at work need to be distinguished between 'push' forces that drove Chinese people out of China, and 'pull' forces that drew them to the United States specifically. Of the roughly 1.5 million Chinese people who emigrated from the coastal provinces between 1840 and 1882, only a third emigrated to the Americas, of whom around half migrated to Latin America (mostly Cuba and Peru), not the United States (or Canada for that matter). The two-thirds who did not migrate to the Americas largely moved within Southeast Asia, and some to Australia. In other words, Anglophone North America was the destination for only one in six Chinese migrants during the mid-19th century emigration wave.
The majority of immigrants to the US in the 1850s did so in connection with the so-called 'coolie trade'. 'Coolie' probably comes from the Hindustani word quli ('slave'), and may have been phono-semantically loaned into Mandarin as 苦力 kuli ('hard (lit. 'bitter') labour'), although the Cantonese term 咕喱 gu lei is clearly a phonetic loaning. The Hindustani and Mandarin etymologies give away what was basically happening: most of the 1.5 million Chinese emigrants between 1840 and 1882 were migrating as labourers bound by contracts (if there were contracts at all), with an at-best nominal difference between ‘coolies’ who worked as indentured labour, mainly in Latin America, and ‘free’ workers, mostly in North America, who were nevertheless effectively bound my multiyear contracts. The ships they travelled on were manned largely by European and/or American crews, typically with a huge disparity in numbers: on the Norway, where the Chinese passengers mutinied in 1859, there were some 60 Europeans to over 1000 Chinese, or a ratio of 1:16. In their construction and organisation, the coolie ships resembled slave ships, with iron gratings and armed guards. Moreover, an uncertain but likely considerable number of labourers had been kidnapped – in 1860, during a crackdown on kidnappers in Whampoa orchestrated by the Qing viceroy of Liagguang, some 29 kidnappers were captured and 77 victims released, and 'scores' more were freed from American ships by the US consul. Apt or not, comparisons to the African slave trade, which had been nominally abolished earlier in the century, were common in the Anglophone press, and raised questions about British and American complicity (or at least acquiescence) in the coolie trade. Public outcry in America, some compassionate but much, undoubtedly, racist, as well as a resurgent Qing government, led to the official abolition of the coolie trade in 1874.
An aspect that may be worth bringing up is that this migration was mostly of men. Thanks to the ultimately labour-centric origin of coolie recruitment and the dominance of the coolie trade in this early phase of migration, women are unlikely to have made up an appreciable portion of Chinese emigrants at all before the emergence of more regularised migration policies in the 1870s. Although Chinese women certainly did emigrate to some extent before then, the numbers were incredibly low. Although, it must be said that the emigration of Chinese women before the Second World War was consistently rather small. At the peak of Sino-American migration in 1929, women made up less than a sixth of total migrants. Alan McKeown, in his 2010 article, Chinese emigration in global context, 1850–1940, offers Chinese family structures as an explanation. Chinese custom favoured the wellness of the broader lineage over the notion of a 'nuclear' family, and it so was normal to expect the temporary or even permanent removal of some male family members to another location to earn money which would be remitted home. When people were emigrating for economic reasons, it made sense within this general cultural mileu for men to dominate. An interesting comparison is made available thanks to McKeown’s data including the 1930s: from 1938 to 1940, when the Japanese invasion of China was underway, women made up closer to 40% of Chinese emigrants, which would fit with the assumption that in wartime, many families were evacuating in their entirety, instead of just the men leaving to support families that stayed behind, as would have been the case in earlier decades.
As your question is specifically on the 1850s I will share that focus and try to keep to the earlier period, but it is worth noting that the coolie trade had commenced back in 1847, when a plantation-owning family in Cuba contracted British merchants to arrange the recruitment (perhaps a euphemistic term) of a group of Chinese workers. That one of these merchants happened to be the Spanish consul in Amoy (it's a long story) meant that there was a pretty obvious conflict of interest between his official duty to guarantee that the 'recruitments' were not made under duress, and his personal profit motive. While not all cases of coolie trading in the coming two and a half decades involved such overt abuse of diplomatic authority by the direct participants, it is hard to avoid the general sense that trans-Pacific coolie transport involved huge amounts of involuntary coercion.
At the same time, such coercion was only part of the picture. To quote Elliott Young’s Alien Nation (2014):
Few Chinese emigrants fit the extreme ends of the spectrum from completely free and voluntary migrants to totally defenseless kidnapped victims. Most found themselves somewhere in the middle, coerced into signing contracts because of debts and hopes of providing for their families.
While there were known cases of young men being indiscriminately kidnapped and forced into coolie work, given the typical promise (in the 1850s at least) of an 8 dollar advance and a minimum of 3 dollars’ pay a month (about 4 to 8 times the pay for a typical unskilled labourer in China itself), a stint as a labourer overseas would have been an attractive prospect on its own, and some did sign on fully-informed. This was not a large proportion: the 1874 Cuba Commission report on the coolie trade concluded that only about 7% of coolies were fully voluntary migrants. Yet at the same time, only about 7% were outright kidnapped. 10% were ‘ensnared’ (forced to become coolie labourers to cover gambling debts), 5% were ‘entrapped’ (believing they were filling in for someone else on a short-term contract), and 72% were ‘decoyed’ (where some other form of fraudulent dealing took place). The vagary of the ‘decoyed’ category is unfortunate, but there is a critical takeaway: not only was there an overall mix of reasons for getting stuck in the coolie trade, most individuals who did get caught up in it did so for mixed reasons. The unscrupulous ‘recruiters’ by and large exploited individuals who were already inclined towards taking on an overseas labour contract of some description.
Young points out that the exercise of agency by coolie labourers went a step further. Many of the mutinies that took place on coolie ships turned out, on investigation, to have been the result of pirates intentionally enrolling onto coolie ships specifically in order to launch mutinies and loot the ships before escaping back to dry land. The exploitative coolie trade was exploited in turn by those willing to game its systems.
I bring all this up partly because it’s interesting, but partly also because it gives additional context to the push-pull factors at work. In particular, there were foreign economic interests and local collaborators doing a lot of coercive ‘pulling’ alongside the more abstract ‘pull’ of overseas opportunities.
Traditionally, the critical ‘push’ factors have been said to be firstly population growth and consequential increasing competition for resources and opportunities in China itself, and secondly the outbreak of massive revolts in China during the 1850s and which reached critical pitch in the early 1860s. A version of this argument can be found in June Mei’s 1979 article, Socioeconomic Origins of Emigration. To quote straight from her conclusion:
Fundamental economic changes caused peasants to be slowly separated from the land; war, foreign competition, and domestic strife made it difficult for many to find alternative work in China; myths of quick riches and active recruitment caused some to turn to California for a livelihood.
The scale and nature of these ‘push’ factors cannot be understated. In Guangdong and Guangxi, for instance, population pressures accentuated inter-ethnic strife, which ballooned into the Hakka-Punti Clan Wars, and drove many to seek opportunities in places that were not simply more prosperous, but indeed far safer than conditions at home. Moreover, in the early 1850s, Guangdong, Guangxi, Zhejiang and Fujian saw the formation of armed secret societies and bandit gangs. A further, empire-wide crisis was the ongoing fallout from the Daoguang Depression, which had seen a massive collapse in the relative value of copper coinage to weighed silver, where by the mid-1840s the nominal ratio of 800 copper coins per tael of silver had tripled to some 2400, and this ratio would never return to pre-Depression levels. More locally, the economy around Canton in particular declined following the opening of new treaty ports at the conclusion of the Opium War, although provincial economies grew overall, even in Guangdong. In short, the south Chinese provinces in the 1850s, particularly the early 1850s, were overpopulated, economically underperforming, and dangerous.