Yes, at some time in the distant future, they'll have to emigrate to survive, if they've survived that long.
But ego remains a potent force against that survival. Ego is self-importance and civilization requires working beyond such action-reaction mentality.
When the sun becomes too hot (long before the sub-giant stage and while it is still well within the main sequence), the Earth will not be able to sustain an ocean. The surface will become far too hot to sustain life.
There may be many star systems nearby which have habitable planets in the right distance and mass, but we still know nothing about the chemistry of their atmospheres. Free oxygen implies photosynthesis and thus life, but will the life be dangerous to us? Perhaps it would be ideal to find a planet the right distance and mass, but with a carbon dioxide atmosphere which can be cooled down and terraformed. But that's a lot of work.