Only one force counts on galactic scales: gravity.
Stars like Alpha Centauri are orbiting in the galaxy, as is the sun. Both stars are moving in a similar direction and at roughly the same speed, relative to the galactic centre. The distance to such stars will vary slowly, but both stars will remain within the galaxy, and the stars will stay fairly near each other for some time yet. The direction of stars doesn't change suddenly, but the gravity of the galaxy will pull the star's path into a roughly circular orbit within the galactic disc.
High-velocity stars are still affected by the gravitational pull of the galaxy. The galactic escape velocity is over 500km/s (1800 ly per million years) Stars can have sufficient speed to escape from the galaxy's gravitational pull. Such stars can escape from the galaxy into intergalactic space. They will be decelerated by the gravity of the galaxy, but if their speed is enough, then gravity won't be enough to pull them back. Stars that are moving more slowly than that will pass into the galactic halo, before being pulled back by the gravity of the galaxy. They will then usually orbit in the galactic halo, with an orbital period measured in hundreds of millions of years.