A swarm-Dyson sphere, which is the only kind that can be realistically built, is incremental. You just keep building habitats in orbit around the sun, until eventually you capture all the sunlight.
Clearly a solid Dyson sphere is not orbiting its host star, and could eventually drift into the star. However, might it be possible to harvest enough energy from the star to power a propulsion system that keeps the sphere in place?
Won't gravity stop this from happening? Seriously, the shell of the sphere will be balanced on the outskirts of a gravity well right? So if it moves this will move the center of the gravity well and both the star and sphere should fall back towards the center.
