The states with the highest percentage of people born in the same state tend to be some of the least dynamic states, either in the Deep South or Rust Belt states with aging populations. Louisiana is currently at 79 percent, followed by Michigan at 77, Ohio at 75, Pennsylvania at 74, and Mississippi at 72. On the other hand, the ones with the lowest percentage of people born in that state are ones that are rapidly diversifying (and, except for Wyoming, turning blue): Nevada at 25, Florida at 36, the District of Columbia at 37, Arizona at 38, and Wyoming at 40.
It might seem a little surprising that California isn't among the lowest (it's at 55), but that's consistent with the sense that California's once-rapid growth has leveled out lately; where California leads the way is percentage of people born outside the USA, at 28. It's followed by New York at 24, New Jersey at 23, and Florida and Nevada at 21. The percentage of foreign-born is only at 17 in Texas and 15 in Arizona, which may partially explain why they're aren't turning as blue as rapidly as people would like them to.
There's one other interesting category I noted: the states that have the largest percentages of people from one other particular state. Leading far and away is New Hampshire, where 25 percent of the population was born in Massachusetts (which explains some of New Hampshire's political shift leftward in recent decades). That's followed by Nevada, where 19 percent of the population was born in California. The third place state will probably surprise you, though: you might think it was Floridians born in New York, but no, 14 percent of the people in Oregon were born in California. (Having grown up in Oregon myself—where resentment of Californians is nearly as legendary as New Hampshire's resentment of "Massholes"—it certainly didn't surprise me).
The New York Times' Nate Cohn seizes on this data to look at the political implications. In particular, he looks at how the rest of the South isn't diversifying as rapidly as Florida, Virginia, and North Carolina—those three states are the only southern states where there's more in-migration from the Northeast and West than from other southern states. That, plus lower migration from outside the United States, partly explains why the other southern states (including Texas, as mentioned above, and Georgia) aren't as likely to follow that trio into swing-state status any time soon.