Doesn't generally happen that way, in my experience.

Example: A friend of mine is married to a Frenchman[1] who came over on (I'm pretty sure) an H-1B, and who was in the process of getting a green card when they started dating.  The process is expensive and a pain in the ass, but it's a pretty boring and formal process.  I don't think his employer cared much either way.

[1] (See, Jerome, we love the French, even in the South.  And he's not even one of those fake Canadian ones.)

Conservatives want live babies so they can raise them to be dead soldiers. - George Carlin

but but isn't that exactly the issue with greencards in the US.

Why, yes, Helen :) picking scabs is an age-old American pasttime. When when labor organizes and agitates and  wage demands squeeze margins, corporates shanghai recruit the world's poor and yearning to be free. By the boat-load.

Oh. That scrubbed Detroit News article, "$10M cost of do-over is obstacle; powerbrokers in Mich. to press solution with national party"? I saved bits I thought interesting back when... March 2008.

The governor, in an interview with The Detroit News, referred to the contest as a "firehouse primary" -- more expansive than a party caucus but not a full-blown affair like a traditional, state-financed primary. People would have to declare themselves Democrats in order to participate, and the contest would be run by the Democratic Party, not the state.
    [...]
In another development, a new Michigan team was set up Thursday to talk with the Democratic National Committee about a resolution. The members are: U.S. Sen. Carl Levin, U.S. Rep. Carolyn Cheeks Kilpatrick, Democratic National Committeewoman Debbie Dingell and UAW President Ron Gettelfinger. All are neutral in the presidential race.
    [...]
The national party needs to ante up or help raise money to pay for it, she said. State taxpayers already shelled out $12 million for the Jan. 15 primary, and Granholm said she won't ask for public funding for a make-up contest. Hillary Clinton won the January primary, but she was the only major candidate [sic] on the ballot. The others, including Barack Obama, had their names removed.
    [...]
Granholm and state Democratic Party Chairman Mark Brewer said the national party has been pushing for party caucuses. Brewer estimated the cost of running a firehouse primary at $10 million, to cover the mechanics, staff and publicity.

There's always more to the story ...

Diversity is the key to economic and political evolution.