Mining is not an instantaneous process. If they slowly mined the mountains of unobtanium (if that IS what makes them float), then the mountains would be sinking gradually, not instantly falling out of the sky without the benefit of their floatyness, because the unobtanium would be mined gradually. So the process of jumping into an airship would be easily accomplishable.
And did you see the movie? I don't think the government would give a flying fuck about the land's damage. They were gassing and burning the natives and their homeland, not to mention the damage to the surrounding woodland.