Immigration is undoing the Mexican-American War of 1846-8, which set most of the current boundary. But boundaries are not impervious, immutable barriers: The Economist points out that in significant ways that border change has been undone.
The Treaty of Guadalupe Hidalgo
ending the war had a dual nature. One was a transfer of territory occupied and controlled by Mexico It was also a quit-claim deed for the vast majority of the territory covered by the agreement in which Mexico had no control.
Likewise, based on its coastal exploration Spain had a claim to the Northwest, which they gave to the USA in 1818 as part of a quit-claim deed: the Adams–Onís Treaty, which also transferred control of Spanish Florida to the USA. What's interesting is that the political results of 1818 and 1848 have been somewhat undone socially, by people voting with their feet. The Adams-Onis treaty transferred actual Spanish-ruled colonies in Florida, but just renounced Spain's unsubstantiated and unrealized claim to the Northwest, of which it had no control. The people who lived in the Northwest, the owners-in-fact, were not a party to the agreement.
The Louisiana Purchase was similar, transferring title to French holdings in New Orleans and St. Louis, but involving mostly unrealized vast French claims in the Mississippi and Missouri drainage without any consideration of the Native owners-in-fact. The 1848 peace treaty with Mexico surrendered title to California missions, Santa Fe, Albuquerque, but again covered mostly unrealized claims to the interior of the Southwest.
The larger issue here is American imperialism, and its denial in American Exceptionalism. If America is a figurative Shining City on a Hill, it is a city much like ancient Rome, and however well Rome's citizens were treated it attacked its neighbors until it ruled the Occident. Imperialism has been widely practiced worldwide for millennia. My point is not that it shouldn't have happened - I don't want to apply today's standards retroactively - but that much of American Exceptionalism is a myth, that America was and is much like any other state. Howard Zinn has a strong critique focusing on American treatment of its people, citizens and non-citizens alike, which I recommend. But I am focusing on nineteenth century American territorial expansion and conquest.
The most exceptional thing about America is that, partly intentionally but mostly through inadvertent disease susceptibility, over 90% of the Natives died after European colonization. This made imperialism, and specifically colonial replacement of the Native cultures, easier than it was in Africa, Asia, or even Mexico (but like Costa Rica). Now we have a lot of company in our imperialism, East and West, and colonialist legacy covers the Western Hemisphere.
Our Manifest Destiny - a term coined in 1845 and used to justify war with Mexico and westward expansion in general - was closely foreshadowed by Ch'ing China expanding west and Imperial Russia expanding east in the 17th century. The Adams-Onis-like Treaty of Nerchinsk in 1689 partitioned East Asia regardless of the owners-in-fact. Chile and Argentina expanded south into Patagonia as America expanded west. We're not that special.
The second most exceptional thing about America was its pioneering experiment in democracy, first for English Protestant male land owners but for increasingly broad segments of the adult population. Again, we have a lot of company in democracy now. What remains exceptional is our military, and (for now) our wealth. China was the richest nation on Earth into the 19th century, and is more likely than not to be that again by the end of this century. We should set aside reverence for an ephemeral, mythologized past and make the best of our all-too-brief time at the top - and remember that we will not always be that. We have been planting some bitter seeds.
America has had unprecedented military and economic strength since World War II - and especially post-Cold War - that has made it unique in global history. That economic strength has declined, relatively: we account for a ¼ share of world GDP now, down from ½ in 1945 but still the highest. Militarily, however, we account for almost half the world's spending and have the greatest armory in the history of the world. Those things have made the scale of what we do novel, but not qualitatively changed it in any distinctive or positive way. We throw our weight around just like any other historically dominant state: for instance, 134 nations have American Special Forces in them.
We are still subject to all the historical forces all nations have been subject to, and are not qualitatively unique. We can get better as a country or we can get worse. There is no standing still. If you would like to extoll America's virtues, of which there are many, I welcome it. My point is that our ethnocentrism is common. We cloak it in, among other things, a mythical exceptionalism for which I have little patience - an exceptionalism smacking of the divine rights of kings coupled with the dangerous and volatile conceit of being God's Chosen.
It is true that people can come here from any.other.country.on.Earth and become American. It can't happen to the same degree in France, China, India, or Turkey - or anywhere else in the Old World. But in the New World, the USA is not alone in that.
Insofar as I believe in American Exceptionalism, it is this: however much we may overstate it at times, we do have an unusual sense of how much we change the world. But even that is more a reflection of our isolation from the Old World in the 19th century followed by our military and economic preeminence in the 20th. When my nation is wrong, it is my duty as a citizen to make it right. As much as our character is tested when we are weak, it is tested more when we are strong.