The secret to moving an asteroid is to realize that it is easier to move half of an asteroid than a whole one. In other words, use some of the asteroid’s mass as rocket fuel to move the rest.
The Case for Ballast shows that ballast is the only adjustable parameter for determining the thrust of a gravity tractor. Ballast, or mass augmentation, means mass added to the spacecraft after launch and mechanically stowed.
Chondrulic Rockets explores many possible ways to use asteroid mass to develop rocket thrust. The lowest energy solution for moving an asteroid is a chondrulic rocket, or mass driver. Most chondrulic rockets require ballast.
You can design your own planetary defense spacecraft using this Grand Challenge spreadsheet.
The video, Deflecting Asteroids tells how a gravity tug can save the Earth by carrying ballast and throwing rocks.