Is free-trade beneficial if living costs are unequal between countries? Free trade, it is said by those who support it, makes the whole world richer. Yet it seems, that if things like minimum wage, taxes, or living costs are unqual between countries then free trade would only be advantageous to the poor countries and to the upper echelons of the rich societies.

For example, a factory in Antarctica would have higher living costs for the workers than a factory in China. Hence if the Antarctic peoples were formerly self sufficient, the upper classes would soon transfer all labour to the cheaper countries and the working classes would starve, and most likely freeze to death.

This seems like what happened in the EU where there was free trade but because the taxes were different in each country, it lead to a disaster.

How is it possible to make free trade work for the benefit of all people?
The classic answers is transfers through welfare programs, re-training etc. See this question which is almost a duplicate. I'm re-quoting from there, just with a different emphasis:


  But just because the size of the pie expands [due to free trade], it doesn’t mean that everyone is better off. There are going to be some losers whose slice of the pie is so much smaller that they would have been better off with less trade. However, because the overall size of the pie has got bigger, the government can compensate the losers which can still make everyone better off. [...] But remember: thanks to free trade, you can afford that [better welfare programs], because the overall size of the pie is bigger.


Or


  Trade enables countries to experience economic growth and a rising standard of living by increasing access to physical capital and export markets. However, not everyone is better off as a result of international trade. Some domestic businesses and industries fail in the face of foreign competition, which results in job losses for workers. Economists suggest, however, that policy solutions that impose trade barriers are harmful to the economy. Rather, they propose policies such as those that provide job training programs to assist those displaced by trade.


(If you want to ask about the Eurozone crisis, please ask separately, I don't want to get into details on that here. Suffice to say that it wasn't simply a matter of free trade, but also of a currency union etc.)