Thomas, you're likely thinking of your Irish context of economic migration of Irish people (Presbyterians in the 1700s, Catholics in the 1800s) out of Ireland due to major cultural, political and economic uproars caused in Ireland by the British government, sometimes not by accident but by design.
Now granted, the Scots-Irish migrations to North America in the late 1600s and the majority of the 1700s were not from British territory to somewhere else, but rather from one part of the realms of the King to another part, all under his royal sovereignty though (depending on the year) not necessarily under the authority of the Parliament. But the Irish emigration of the 1800s to the United States was different -- people were renouncing (perhaps more accurately, fleeing) the rule of the British and coming to a country which had fought a war to be free from the rule of the British.
I don't see any biblical ground for that being a problem. The United States needed immigrant labor (though it often treated immigrants terribly). The British wanted to get rid of its "Irish Problem" and promoting emigration helped encourage potential "troublemakers" as well as poor people to go away. It seems like a win-win for the Americans, for the British, and for the Irish emigrants themselves.
The problem is that the United States today is not the United States of the 1800s. We want to have greater control over who we let into the country because we don't need large numbers of low-wage workers (though we do need some). What we need is skilled workers and professionals.
Mexico and a number of Latin American countries may want to get rid of their problems by sending them to the United States, but we don't want illegal aliens who break the law to get here.
My Italian ancestors followed the law to come to the United States. My uncle recently sent me the Italian birth certificate and American naturalization papers of the first of my ancestors to become an American citizen. He became a citizen about three decades after he came to the USA. He came the right way, legally. My Korean wife did the same thing, coming here as a graduate student with no plans to stay, but meeting and marrying me, not to get American citizenship -- she obtained a green card based on our marriage but kept her South Korean citizenship for many years and actually didn't become a citizen until long after she was eligible to naturalize -- but because we met and married. If I had been a Canadian or Australian or British citizen attending graduate school in the United States, she would have gone with me to that county, not stayed in the US.
I think the key biblical issue here is obedience to Romans 13.
If I want to live in South Korea, I have to follow the laws to go there. Their laws are strict and it is not easy for a foreigner to get permission to live in South Korea. If I wanted to live in Italy, until a recent change in Italian law I could have applied for Italian citizenship based on my Italian ancestry by right of blood descent, but that's not something every country does. (I think there's a similar "jus sanguinis" law for Ireland but you'll know better than me.)
I have no right to just cross a border and move somewhere because I feel like it. Mexico won't let me do that in their country. Why should we let Mexicans (or others) do it in our country?
This has nothing to do with being anti-immigrant or anti-minority. There was a day when Italians were viewed as the "lowest of the low" among immigrants to the US, and tolerated only because we would "do work Americans won't do." In other words, the same socioeconomic bottom rung now filled by Hispanics, and for many of the same reasons. The last thing I want to do is exclude people based on being Hispanic, and I've actually been saying for decades that we need to create a legal immigration pathway for agricultural workers, an area of low-skilled or unskilled labor for which we really DO need immigrants in the American Southwest. Decades ago we had such laws and we probably need to re-establish them, rather than our current system in which we ban such immigration but then play wink-and-nudge games because farmers really DO need labor to pick the crops, so there's an economic incentive to break the law.
Biblically, what counts is whether a country wants to let me move there. It's their country. Not mine. Their government sets the rules. If I break them, it's no different from me walking into your house without permission and setting up a tent in your living room and deciding to live there without your permission.
"Render unto Caesar" applies. And in the late 300s and early-to-mid 400s, the Caesars found out the hard way what happened when they allowed Germanic tribes to cross the borders, come into the Empire, and start following their own laws and customs without regard to Roman law. Initially they asked permission. Later they came whether they were wanted or not. Eventually they destroyed the Roman Empire.
That's what happens when governments don't, or can't, police their borders.
