Of course people have a right to work in any profession they choose. People ought to be able to do whatever they want. They do not, however, have a right to get paid for it.
Take your mining example. South African coal is only cheaper than English coal because South African miners get paid a lot less. If English coal miners want to take a pay cut of 50%, I’m certain that English coal would be extremely competitive. They don’t. So the real question is, do people have a right to work in a specific profession at a specific wage?
Of course not. Turn the question around. Do I have a duty to give money to someone so that they can work in a specific profession at a specific wage?
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Things like logging, at least in North America, are a bit different. Take the controversy surrounding logging in old-growth forests. “Stop!” cry the environmentalists, “Old-growth forest is a valuable environmental resource. It will all be gone in twenty years if we keep logging it!” “Listen, you tree-hugging freaks!” reply the loggers, “People are more important than owls. You are trying to destroy our communities and our way of life!”
On this one, the environmentalists definitely have the better of the argument. These communities and their way of life will disappear in any case, once the resource on which they depend has been consumed. Since they’re going to take the hit eventually anyway, we might as well get it over with now rather than wait twenty years when there will be no forest and no communities.