I'll answer your question, framed, but not in the misleading terms in which it is put. This is, as you have said in relation to the 19th century Dundee Jute industry, hardly a new issue. Thoughout the development of capitalism the capitalist class have brought in new labour supplies to feed their boom periods, and of course undermine the ability of the indigenous workers to use basic restriction of labour supply tactics to force wage rates up. The biggest single cohort of migrant workers brought to Britain for this purpose historically was of course the Irish - prior to the major influxes of Commonwealth and now EU migrant workers much more recently. Who benefits from an unlimited labour supply ? The capitalist class of course, certainly, for one. Though the downstream benefits to the migrant worker receiving economy and population generally are considerable too - being mainly young and fit and already trained the welfare costs to the economy of this new labour pool is minimal. Also a lot of migrant workers do jobs (such as meat and poultry production) which is so horrible that indigenous workers simply don't want to do it. Therefore the entire society, and living standards for everyone in that prospering economy, can be argued to benefit from the cheap labour inputs of migrant workers. It is also an issue of considerable academic debate as to whether the influx of new labour supply has been a major factor in the falling share of national income going to ordinary workers wages over the last 30 years. Other factors, eg, trades union restricting legal impositions and broad work restructuring (outsourcing - breaking up the "big battalions" of labour), globalisation of production (shifting Big Industry to the oppressive states of the former Third World), and saturation by the current overwhelming mass media "free enterprise" pro capitalist ideological hegemony , are all more important elements in the toxic package that is the neoliberal economic system.
Nevertheless an unlimited new labour supply does undoubtedly does make it harder to force up wage rates by traditional trades union labour supply restriction alone. Does this mean the Left and Labour Movement should therefore campaign to restrict immigration though ? Here one has to step well back from the simplistic and poisonous idea that campaigning to restrict immigration would achieve anything positive for the working class at all. Firstly the capitalist class aint going to agree to restrict free labour entry - so a huge campaign would have to be mounted - based on the completely false premise that incoming migrant workers are a key element in the indigenous population's current economic hardships. They aren't a key though cause are they ? The bankers crashed the world economic system , not migrant workers. The Falling Rate of Profit since the 1970's and the chaotic nature of the capitalist system generally, produces its crisis - not migrant workers. The Austerity Offensive is a major offensive against ALL workers of whatever origin, by the capitalist class, worldwide. To redirect "indigenous" worker's anger in each state at their declining living conditions to their fellow workers of foreign origin - is to play directly into the divide and rule game of the bosses .
Given the Catholic Irish origins of many IWCA activists, I would have thought at least the folk memory of the vicious scapegoating of Irish migrant workers across Britain during the late 19th and early 20th centuries by racists suggesting "if only the Irish would fuck off back to Ireland and stop competing for our jobs - we'd all be better off", would have put off the IWCA from falling into this completely diversionery and reactionery political position. Apparently not. The IWCA, and obviously you too , framed, are all too ready to fall into the same old divide and rule trap. During the 20th century there have been a number of attempts by confused workers, particularly in the USA, to set up trades unions which would extend further the entry restrictions of Craft Unionism - to entry only by racial or national identity - culminating in that famous reactionery modification of Marx's old slogan - to become "White Workers of the World Unite".
All the IWCA "analysis" and "critiques" of the Lefts' positions on a range of issues , "multiculturalism/identity politics", immigration, child abuse, drug dealing, Islam, actually does is pander to the prejudices of the most politically backward sections of the poorer, least educated, least organised , sections of the working class - the "Lumpen Proletariat" in fact - the historical source of minor "proletarian" support for the largely lower middle class composed fascist movements . There is nothing new about pandering to racism or cultural prejudice for electoral benefit by supposedly "Leftish" groups, or claiming that it is" pro-working class" campaigning to restrict immigration. This line of politics leads those who hold to it eventually to some form of working class neo fascist "Strasserism" - and of course to eventually siding with ones "own" ruling class and "national interest" against the workers and "their" bosses of other nation states. It just does. Always. Any organisation which goes down that route is doing the work of the bosses in dividing the working class and confusing them as to who the priority real enemy is - the capitalist class and their capitalist system.
You really are stupid, arent you? If you agree with that statement, then you agree that everything YOU are arguing also supports capital.
It's almost nostalgic, this tripe. 
