Moving mass back with a free return trajectory is much easier than getting stuff out of Earths gravity well and up there. Escape velocity from an asteroid is lo

Moving mass back with a free return trajectory is much easier than getting stuff out of Earths gravity well and up there. Escape velocity from an asteroid is low and if you yeet it with enough precision you can send it on a trajectory that will land it on Earth. As for moving the asteroid you could use extremely high specific impulse ion thrusters, in situ propellant from the asteroid itself, or a solar sail. But you are right that simply firing the valuable stuff back to Earth is probably easier.