Gangs of New York has many historic inaccuracies, and this scene in particular is one of them. First, it perpetuates a popular myth that immigrants, especially Irish Catholic immigrants, were the bulk of rank and file in the Union army. To quote James McPherson:
"The substantial number of immigrants in the Union army gave rise to longstanding southern myth that "the majority of Yankee soldiers were foreign hirelings." But in fact quite the opposite was true. Immigrants were proportionately under-represented in the Union's armed services. Of some two million white soldiers and sailors, half a million had been born abroad. While immigrants therefore constituted 25 percent of the servicemen, 30 of the males of military age in the Union states were foreign-born. Despite the fighting reputation of the Irish Brigade, the Irish were the most underrepresented group in proportion to population, followed by German Catholics. Other immigrant groups enlisted in rough proportion to their share of the population."
McPherson notes that those who enlisted or were drafted into the Union army tended to be disproportionately Protestant and skilled laborers or farmers, not working class Catholic immigrants. Which would make sense because especially in a place like New York City the latter groups tended to be extremely loyal to the Democratic Party and suspicious of a war effort largely run by Republicans such as Lincoln and Seward. Indeed, it was the Democratic Party that promoted the "rich man's war, poor man's fight" slogan in political debates over passing the 1863 Draft Bill (the passage of which led to the July 1863 Draft Riots in New York City, which are depicted in a historically distorted and whitewashed form in the film). For what it's worth, according to McPherson 98 percent of the men drafted in heavily Irish-immigrant districts ended up paying commutation fees or hired substitutes (mostly though Democratic political machine Tammany Hall funds).
That's some background. I slightly sidestepped the question as to whether the scene in the film actually happened. It seems like Scorsese is depicting the scene mostly as a metaphor for the myth mentioned above, ie, that immigrants "fresh off the boat" were signed up to serve in a meat grinder (hence the coffins being unloaded right next to the fresh volunteers), so I don't think the viewer is supposed to take the scene as the literal truth.
I've downplayed the role that Irish immigrants played relatively speaking in the Union army, but (as McPherson himself admits), there were notable exceptions, most notably the Irish Brigade, raised in New York by Capt. Thomas Francis Meagher and Corcoran's Legion formed by Colonel Michael Corcoran (both were Irish immigrants and connected to the Fenian movement).
It looks like the recruiting scene is a reference to recruiters outside Castle Clinton: the National Parks Service said that recruiters were located here to recruit recent Irish immigrants, but they are vague as to if this was before the Civil War, or before.
However, Irish historian and archaeologist Damian Shiels (who wrote The Irish in the American Civil War) looked at the recruiting in this area and found a letter where the New York Commissioners for Emigration explicitly banned the New York County Volunteers Recruiting Committee from recruiting in Castle Garden (specifically because it would spread the idea that new immigrants were being duped into joining the Union Army, and the bad international press this could cause).
Shiels' takeaway is therefore:
"The more I investigate the Irish experience, the more apparent it is that the type of incident portrayed in Gangs of New York rarely, if ever, occurred. Far from being duped, it was much more likely that many of these men had travelled to the United States with the express intention of joining the military, in the hope of benefiting from the financial rewards available for doing so. This was the primary motivation for Irish enlistment in the Union Army from at least 1863 onwards. These men were not stupid- they came from a country where enlistment in the British Army for economic reasons was commonplace, and they came informed about the Civil War."
Sources:
James McPherson. Battle Cry of Freedom: The Civil War Era
National Park Service. "Irish Soldiers in the Union Army"
Damian Shiels. "Gangs of New York: Recruiting the Irish 'Straight Off the Boat'". From his website.
Well, apparently the U.S. was doing some covert recruiting in England and Ireland in addition to the above-board encouragement of immigration from western Europe. Just, never quite enough to make the British anything more than vaguely suspicious. Meanwhile, on American soil, British consuls informed British subjects of an 1861 law that barred them from serving in either side's army. Just, never quite loudly enough to make the Americans anything more than vaguely suspicious.
Well, except for that time in Philadelphia and that other time in Chicago, but I digress.
You can read about the (lack of) historicity of that particular scene a bunch of places online, so I'll address your actual questions: (1) how easily were immigrants conscripted (2) potential effects on immigration.
As the Civil War ground on, both the U.S. and Confederacy had an increasingly dire need for more soldiers. One of the ways that both sides sought to accomplish this goal was by lowering and lowering the bar of who counted as enough of a citizen to serve, and eventually, to be conscripted. That effort had to be balanced with potential diplomatic costs, since they also each needed trading partners on their side.
Traditionally, military membership in the U.S. was done at the state level, not national (something something "Confederate" flag something). Well, because the Civil War was absolutely, 100% about states' rights and the tyranny of a federal government (narrator: it was, in fact, not about states' rights), Confederate president Jefferson Davis signed North America's first national-level conscription act in 1862. It applied to "white men who are residents of the Confederate States." Well, that's pretty ambiguous as to who counts as a "resident."
Indeed, foreign consuls--especially the British and French--were pulling the equivalent of all-nighters before finals week to issue certificates of exemption to their country's citizens. (Or to naturalized American citizens who decided to self-issue an unnaturalization.) And apparently, the Confederacy was mostly willing to honor that, even to the extent of discharging nonnaturalized immigrants if they were conscripted by accident. Well, or by kidnapping, which may or may not have actually happened but got a lot of press for doing so.
As you might expect, mounting casualties and ongoing war made the Confederacy decreasingly likely to accept certificates of exemption.
The United States likewise faced the problem of who counted as "enough of" an American to serve. As late as 1862, Secretary of State William Seward was promising that only full American citizens holding U.S. passports would be conscripted. Certifications of exemption from state-level conscription were in high demand here, too. The New York consul noted wryly that almost overnight, Irish immigrants went from desperate to shed their British citizenship to the proudest British citizens in the world.
The turning point in the U.S. was the Conscription Act (or Enrollment Act) passed in spring 1863, which changed the bar from "citizen" to "having declared an oath to of intention to seek citizenship someday." It automatically applied to older and recent immigrants who had already done so, and apparently some new arrivals were trapped by it as well. A follow-up provision mandated that men who had declared intent and didn't want to serve had a grand total of 65 days to leave the U.S.
Other factors aligned to make immigrants particularly vulnerable. The Act had built in an exemption: if you could pay someone to take your spot or pay the government $300, you could skate on your merry way. Poorer workers--immigrant and native alike--were the ones unable to pay. And who among them could afford to pack up their lives and families and pay the costs of moving to another country? In Ohio, trying to meet its state-level quota after the 1862 Militia Act, potential draft dodgers might be arrested. People complained to their pastors and priests that bosses were closing factories and other jobs temporarily--just temporarily enough to put them in such need of money they'd enlist to serve. On the other hand, media stories about Irish immigrants resorting to desperate measure to avoid conscription were less news than xenophobic propaganda.
Whatever problems arose at home, the U.S. did a fair enough job balancing their diplomatic and military needs. Their own consuls abroad even recruited new immigrants by talking up all the jobs that were going to be waiting for them in the South once the war was won and slavery was abolished. Meanwhile, the U.S. government dangled immediate citizenship upon honorable discharge from the military as a positive rather than negative recruitment tool. The Civil War didn't have an effect on immigration so much as it did on citizenship.
Further Reading:
Similarly, what was the immigration policy like for the confederacy?
Here's the scene in question: https://youtu.be/ns-qtoxnAS8?t=61
What was the process like to obtain US citizenship just before, during, and after the war? For both men and women.
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