Nathan Smith has a very thoughtful speculative essay on that topic. Here is one interesting bit of many:
I would tentatively envision the US experience under open borders as resembling the British and Roman cases, inasmuch as the protocols and ideals of the US polity, as well as its merely ethnic characteristics, would persist in attenuated form, but governing a much larger population would necessitate improvisational and sometimes authoritarian expedients that would cumulatively transform the polity into something quite different, even as it claimed descent from the historic constitutional polity of the United States as we know it. The illusion of continuity would deceive the subjects of the new polity, native-born and immigrant, to a considerable extent, though on the other hand there would be a good deal of lamentation and triumphalism, and only after several generations would historians be able to look back and assess the bewildering transformation in a sober, balanced way.
Certain American ideals would die of their own increasing impracticality, e.g., “equality of opportunity,” the social safety net, one person, one vote, or non-discrimination in employment. Americans might continue to feel that these ideals were right long after they had ceased to be practiced, as the Romans seemed to feel that Rome ought to be governed by its Senate long after real governance had passed to the emperors. I don’t see how public schools could adapt to a far larger and more diverse student body.
I think the most wild-eyed predictions of the open borders optimists will come true, and to spare, but I think a lot of the forebodings of the grimmest open border pessimists will also prove more than justified.
The article is interesting throughout, do read the whole thing.