The advanced civilization in my story is trying to reach as many galaxies as possible in the next few billions years before the expansion of space has sent those galaxies permanently out of reach without some form of FTL travel which my story does not have.
I plan to have von Neumann probe-like self replicating machines powered by laser relays from Nicoll-Dyson beams, possibly a double beam from opposite sides keeping the star in its location whilst sending probes in opposite directions.
Once in the new galaxies the self replicating machines will gather materials and turn the majority of stars into Shkadov thrusters, the collective thrust of the stars and the gravitational hold of the galaxy will move the whole galaxy towards the civilizations home galaxy. This assumption that whole galaxies can be moved was taken from the following videos: https://www.gregschool.org/gregschoollessons/2018/6/12/shkadov-thrusters https://www.youtube.com/watch?v=_VetAm7fCS0
My question is what percentage of the stars need to become Shkadov thrusters to move the whole galaxy? Would there become a point where converting more stars is taking more time than it is worth for the additional speed? The civilization isn't worried about the time the journeys will take, as long as the galaxies eventually arrive to their area of space.