Only if their country of origin is meritocratic and market-driven. As many flaws as the U.S. has, it generally ranks pretty well on these aspects.
Put another way, many societies have artificial ceilings on the social status and income you can attain regardless of your intelligence, education, drive, ambition, and resourcefulness. Caste systems, guanxi, racial prejudice, endemic corruption, drug cartels that will kill you if you don't help them out, organized crime, warlords that will kill you, lack of infrastructure, etc. When you leave that country and go to one that allows free contracts between people, your social standing - and your children's, to the extent that these traits are heritable - should rise to its natural level, where it would be in the absence of arbitrary restrictions.
There's another level of selection bias where people who feel like all these arbitrary restrictions work in their favor don't tend to leave their home countries. So those people who you speak of who would mean revert - the folks that are above average and have a good deal in their home country - would be underrepresented within the immigrant sample, to the extent that they can realize when they have a good deal. (This is the same principle by which contracts and free trade raise general welfare: people who already have what they need choose not to trade and drop out of the sample, which then consists only of people who feel they are better off trading for something someone else has.)
But wouldn't those artificial ceilings also prevent immigration applicants from having the "paperwork" to show their capability which is precisely what the American immigration bureaucracy will look at? It's not like they're letting immigrants in based on an IQ test, they're letting people in because they have elite degrees, are admitted to American universities, or have American job offers. All of those are indicators of status already achieved rather than potential status which has been supressed by local conditions in their country of origin.
In other words, immigrants coming in through the highly qualified routes are already high status in their countries of origin. Indians coming to the US on H1Bs for instance are overwhelmingly from high caste social groups and educated at the best Indian universities.
Put another way, many societies have artificial ceilings on the social status and income you can attain regardless of your intelligence, education, drive, ambition, and resourcefulness. Caste systems, guanxi, racial prejudice, endemic corruption, drug cartels that will kill you if you don't help them out, organized crime, warlords that will kill you, lack of infrastructure, etc. When you leave that country and go to one that allows free contracts between people, your social standing - and your children's, to the extent that these traits are heritable - should rise to its natural level, where it would be in the absence of arbitrary restrictions.
There's another level of selection bias where people who feel like all these arbitrary restrictions work in their favor don't tend to leave their home countries. So those people who you speak of who would mean revert - the folks that are above average and have a good deal in their home country - would be underrepresented within the immigrant sample, to the extent that they can realize when they have a good deal. (This is the same principle by which contracts and free trade raise general welfare: people who already have what they need choose not to trade and drop out of the sample, which then consists only of people who feel they are better off trading for something someone else has.)