There would certainly be issues with uncontrolled migration, coupled with high levels of benefit. It would be inviting people with nothing to 'come and have some free money' with the inevitable consequences.
We do not have this situation.
Everyone in the whole of the EU has the right to jump on a boat/plane and come to the UK tomorrow. That's over 700M people. They have had this right since the Maastricht treaty of 1992 was enacted. Compared to this, the level of immigration is trivial - net migration to the UK in 2015 from the EU was 180,000 which, because of the way we calculate the figures, includes all the students coming here purely to pay to study. In the same year there were 698,000 births cf. 530,000 deaths so net growth of 168,000. All figures from the ONS. Notably of course, there is a right to work, not a right to benefits. Jobseekers' allowance can only be claimed for up to six months, after a residency of three months, and there is no right to housing benefit (source gov.uk website).
Comparing the above facts to what is often presented, I am suggesting that there is a lot of mass hysteria and inflammatory language, whipped up by the press because it makes for a good story. It is very easy to write the politics of hate. It is also very easy to take two things and write that 'the link between them has not been disproved' to create a conspiracy theory (I really wouldn't be surprised to see a headline one day that 'Immigrants Killed Princess Diana' and I know which paper I would be expecting to see it in) but that does not mean these stories are true, it just means it sells newspapers; the trouble is though that people seem to swallow them.
There are definitely financial issues on a grand scale and people at the bottom are not going forward - in fact they are slipping back. Immigrants make soft targets and it is so much more palatable than taking any responsibility yourself, isn't it. Ironically, in the US (which is after all what this thread is about) pretty much the entire population is made up of relatively recent immigrants.
It is very easy to state that 'immigrants are taking our jobs' or doing them for less money, but there is very little evidence of this beyond 'a man down the pub says so'. We have a minimum wage and we live in a free market economy based on the principles of supply and demand. There is also the practical point that simply removing the person doing the role would not necessarily result in someone else getting the position, particularly if the cost went up. Take the construction sector. Assume for argument's sake that building a house takes around 500 man-days on site. Assume that the lack of alternative labour increases the hourly rate by £5/hr. That puts £20,000 on the build price of a house, in labour alone, let alone the increased cost of all the components, deliveries to site etc, pretty much anything with a labour component. Put that much on the price of a house and what happens - does the price stay fixed by reducing profit? No, of course not, the price goes up accordingly but the developers just delay projects like they did in 2008 and fewer houses get built.
What I do find interesting is that what is being advocated is protectionism in the interest of workers' rights, which is a hard line socialist policy of the left. This is pretty much the policy of Momentum and was the original policy of the National Socialist German Workers Party, better known latterly as the Nazi party. Sometimes the difference between extreme right and extreme left actually does complete the circle.
Alec