Yes and isn't it interesting that immigration is left versus right only by country and time period. Many right-wing argue for cheap labor from poorer countries even if it means illegal immigrants. Many people funding right-wing use that labor so they don't have to pay the locals as much and it's a good reason to drive wages and costs down. For similar reasons some left-wing are opposed to it.
In my country we had two communist parties (with around 0% support mind you), one was a workers party arguing for immigration limits and the other was for open borders.
I never really saw immigration, in and of itself, as a right vs. left issue. As you say, it depends on the country and time period, as well as the motives for immigrating and the country of origin. A defector from a communist country was generally welcomed with open arms by the right-wing. When the freedom flotilla arrived from Cuba, they were allowed to enter.
As long as it makes US adversaries look bad, then immigration is okay from a right-wing point of view.
On the other hand, when they come from countries which are considered US allies (Mexico, Guatemala, El Salvador, etc.), then it's looked at differently. Those countries are US allies and considered capitalist paradises, so why would anyone in their right mind want to leave such a wonderful place? That's why the right-wing tends to paint them as they do, since they can't possibly be escaping persecution or oppression - since that would make US allies look bad, and it would make US foreign policy appear disingenuous. Can't have any of that.