One way to move a star, if you have tons of time, is to start with a small- to medium-sized planet, say a Mercury or Venus, close-in to a star.

Adjust rotation (and maybe rearrange some mass) to achieve tidal lock. Then build your thrusters at the permanent subsolar point--nozzles pointing directly at the perpetual noontime sun.

Then you'd fire the thrusters for part of the planet's orbit, when the resulting vector through the sun is in the general direction you want to go.

It'd take a loooong time, and it'd have a subtle effect on the motion of the star across an observer's sky. It'd acquire a wobble, and from some angles it would look like the wobble induced by a much larger planet because of the thrust vector added to the gravitational pull of the planet.

Kind of like some of the stars we've observed that have wobbles suggesting a very large planet circling very close. Ya don't think...? ;-)