If we had the technology to move a planet, we would also have the technology to build a planet-sized space station, which would be a much more efficient use of resources. You could use spin gravity to make the entire mass inhabitable and useful, instead of almost all of the mass sitting in the core/mantle.
We already have the technology to move a planet; it's just orbiting rockets. (Ion thrusters are especially promising here because of the small amount of mass you lose from the planet in the process.) We just don't have the necessary industrial scale to supply enough rockets and energy. A planet-sized space station is almost certainly possible with carbon nanotube ropes, but those have not yet been demonstrated to work in practice.
However, smaller O'Neill-cylinder space stations are feasible even with just steel cables, and I look forward to a future where the vast majority of inhabited land area is in such contraptions. It will take at least 30 years, probably more like 300. The danger is that we collectively take a more destructive course.