What usually happens in such big companies is that you might have to work in another country for the same company of a couple of years. Big but is that you go there as an employee of the source country taking with you the respective salary and not that of the destination country. They usually pay for the fixed expenses, eg. accommodation and even might have an allowance for the daily expenses but depends on the company and still it might not be enough to make it with your salary.
Also you don't really have an option to decline, you are up next then for dismiss and in many countries it is not that easy to find a similar job etc. etc.
Also from friends working for other such companies I have the same input.
Please note that here we are talking about temporary move (a few years) and then you have to go back to your originating country. Furthermore in order to compensate for the increased costs as I mentioned they usually pay some of the fixed costs like accommodation BUT your salary remains the same. Back in the old days (eg. 10-15years ago) they could even pay extra money on top of your salary.
Also keep in mind that the world is not just the US or India, there are many big companies outside of the US and one last point, this works very well the other ways around. People from a country with higher salary to go work for a few years on a country with lower salary so they make money + get extra promotion points for having worked outside of their main country
The key point here is it being her choice to stay. The reason a company can "get away" with doing this is that people are willing to do it.