Some sufficiently advanced species has decided they will move planets.
Let's assume earth-like, life sustaining worlds. The only real way to do this as far as I know is using "gravity tugs" or messing with orbital trajectories (slingshot), again using gravity.
They want this done in a reasonable amount of time (not millennia).
So they build a ship, with the front being a giant ball of something sufficiently massive (and super dense to save space) to have the required gravity. So how small can we make a ship massive enough to move worlds?
You can address the feasibility, rarity of material, speeds and times, all you want but the Question at hand here is the size of the ship. I don't have any real specifics to share here. the galaxy is huge and they might want all kinds of planets from all over the place. Don't break the planet and get it there within the next thousand years or so is all that really matters.
Ideally the gravity of the ship is enough to select a planet to move without disrupting the entire solar system, so not as massive as a star obviously.