So, I envision a world that has forgotten it's en route to another galaxy -- it's been in transit for only a million years, hasn't even properly left its "home" behind (still stars in the sky, at least behind), but while a million years isn't very long as sublight intergalactic travel goes, it's a very long time for a civilization, even fairly long for a species.
The mode of travel is by Shkadov thruster -- take your star and at least the inner part of your solar system with you, accelerating continuously but at micro-G levels. So, a civilization (Kardashiev level 1? 2?) builds the enormous light-pressure-levitated reflector that makes the star itself into a photon thruster, and over the lifetime of a star, the star can travel (close) intergalactic distances.
Doing this with long-lived stars like G, K, and M class is very slow, but these stars, if reasonably young at the start of the journey, have the potential to live long enough to complete the journey, perhaps even with a turnover maneuver (swap the reflector to the other side of the star, to reverse thrust) to arrive in the destination galaxy at a velocity that will capture, instead of just blasting through on a one-way trip to the next galaxy cluster.
A light pressure levitated parabolic reflector, however, will not be passively stable -- instead, it will (like a ringworld or rigid Dyson sphere) require regular or continuous adjustment to stay in position, so the system doesn't veer off course or the reflector tilt enough to just fall into the star (that would be a bad year -- or century -- for all involved).
I'm no astrophysicist, nor even the ordinary earthbound variety -- I took physics in high school and college, but the non-calculus flavor. It's been forty years since I evaluated an integral, and nothing I've read about Shkadov thrusters suggests this -- but there have been laser pulse propelled technology demonstrators that were passively stable on the pressure wave from air expanded by the focused pulse.
So, is it possible to design a Shkadov thruster's reflector with a shape that would make it passively stable, allowing the "passengers" to continue obliviously on their journey (obviously without a turnover to decelerate)