A commenter again objects to the idea that "left" and "right" is a useful categorisation of political ideas.
On the subject of "left" and "right", there is confusion because I use the terms in two distinct but related senses.
When I talk about the long-term political trends--"leftward drift" and so on, what I am talking about is what is sometimes called "progressivism". It would be good to define that more satisfactorily, but it is an intellectual-political movement of great age, oriented around a cluster of ideals mostly centred on "equality". There is no corresponding long-term definition of "right", there is only occasional opposition to progressivism.
In day to day politics, "left" and "right" have much broader meanings, relating to every area of political controversy. Mostly they have no permanent meaning in relation to ideas about policy, and no meaning in relation to practice of any activity outside of politics. They are, however, an essential feature of any kind of struggle for power. There cannot normally be more than two coalitions seriously engaged in seeking power; if you bring any desire to a power-struggle, it is necessary first to get one or other of the competing coalitions to agree to your desire.
The reason these two very different meanings of "left" get confused is because, given the inevitable division of politics into two factions at any given place and time, we label the faction more in line with "progressivism" as "left" and the other as "right". In some cases, the choice is rather difficult and ends up being pretty much arbitrary.
So, immigration is a progressive and therefore long-term "left" demand when it is premised on the equality of "natives" and "foreigners". However, while that is in line with long-term progressive principles, it is actually very new for it to be advanced with significant powerful backing on the basis of those principles. As the commentator rightly observed, going back only a few decades, it was much more a practical issue pushed by businesses in order to advance their own economic interests. Because those businessmen were part of the (short-term) "right" coalition, immigration at that point was more a "right"-wing than "left"-wing demand. It is by no means the only issue falling into that category; for a century the progressive agenda focused on advancing the status of the working class relative to the employing class (because equality is the central progressive value), and so many people define "left" and "right" as permanent ideas relating to the two sides of that economic divide. As the same commentator put it in 2011:
In the world I was brought up in (and you were born into) Right/Left politics was quite simple. At the extreme of the Right there were bosses and millionaires, and the extreme of the Left there were deep-sea fishermen and coalminers...
But during the 70s the world as I knew it changed into something else. The first inkling of descent into (what appeared to me to be) silliness was called “Rock against Racism”. Then there was the Feminist movement, relying on a series of absurd illogicalities and parodying Marxist class dialectics. Together, and with other ingredients, they formed the basis for the time-wasting activities of so many “equal opportunities” employers today.
It is readily observable that that "fishermen and coalminers" model does not hold in the twenty-first century. Indeed, there is massive opposition to Trump from the existing "right" coalition on the grounds that his stated platform is not "right" at all in twentieth-century terms.
The existing right coalition in the United States, however, is still defining itself in opposition to the left coalition on a field of issues which the left coalition sees as essentially done with or for other reasons not currently important. The force of the left coalition is directed in new, but still progressive directions, including open borders, but the right coalition has the habit of not opposing those policy demands. Hence the "Alt-Right", which is the term for opposition to the new progressive demands of the left coalition, rather than opposition to the old progressive demands of the left coalition.
If the Alt-Right takes over the right coalition, we could conceivably get to a situation where the right coalition is focused on policies of advancing the status of the white working class against the white elite, while the left coalition is focused on advancing the status of immigrants against the white working class. Since both of those are actually progressive values, in terms of long-term advancing of equality, it would be one of those situations where the labelling of the coalitions as left and right could be argued to be backwards.
The summary of my prediction in the original post is that that will not happen. I expect the left coalition to back-pedal on immigration, which it only seized on because the right coalition was failing to oppose it..
Another way of putting my prediction is that over the long term "left" and "right" do usually describe politics well (though they aren't guaranteed to), and that the current left demand for open borders is an aberration that will be corrected before it is allowed to destroy a coherent progressive left coalition. It is reasonably progressive to say that foreigners should have the same rights as natives, but it is not practical for the more progressive coalition to actually go and do it.
A fuller historical explanation of the "descent into silliness" is needed as a matter of the first importance. Did the cause of further advancement of the proletariat run into diminishing returns? Was it sabotaged by clever rightists? Was the obsession of the left coalition with that one issue over such a long period of time itself the aberration, perhaps caused by the Russian Revolution and the resulting alignment with Marx?