This.
Done correctly, you could make a continuous loop of rocks around your target planet and one, or preferably two other objects, such as the Sun and Jupiter. The momentum transfer would be small, but it could be continuously repeated.
See this essay by the late Paul Birch
It’s so obvious even a kid could do it - you simply lower the local gravitational constant around the planet and it will move itself. Note, the kids who can do this are in the Q Continuum.
A World Out of Time; the gas giant used was Uranus. As I recall the fusion engine was a “double ended candle” model, one drive flare facing to space and pushing the engine against the planet, the other holding the engine up against it’s own thrust. It floated when not in operation.
It’s relatively simple. You encase a star in a shell and when you open a vent on one side the star moves in the opposite direction. It’s like letting the air out of a balloon. The planet will naturally move along with the star.
To earn a Class 3 Planet Juggler certification one must either make all the planets in a system switch places or reverse their orbits without losing so much as a moon. It’s a common enough sport among the elite of the galactic core, but I warn you the insurance premiums are out of this world.
You’re all thinking too high tech. We could do it today.
Just get all the people on one side of the earth to jump up and down at the same time. Then twelve hours later, get all the people on the other side of the earth to do so. Alternate every twelve hours. It would only take a second out of everybody’s day.
Sure, it’ll be gradual, but so are all the other methods. And this one is far cheaper, too.
Stellar engines have been theorized, although they’d need to be more elaborate than that. For one thing, building that shell would be the hard way of doing it, even it it’s possible.
Archimedes had this one solved 2300 years ago.
We could throw rocks.
For very large values of “simple.”
I defer to the level of expertise disclosed in your username.
In other words, if you’re seriously thinking about moving the entire Earth, you’re blowing gas out of Uranus.