The asteroids can be accelerated and de-accelerated depending on which side of a much larger rotating object the asteroid passes.
If asteroid passes on the side in the direction of the rotation of the object, the asteroid experiences a gravity boost.
If the asteroid passes on the side opposite to the direction of rotation, the asteroid experiences gravity breaking.
If the breaking slows down the asteroid sufficiently so it can be captured then it collides with object.
An example of gravity assisted flight, the Mars probes used the gravity assist of the Earth to reach Venus, then the gravity assist of Venus to fling it towards Mar. Landing on Mars used gravity breaking.
But for the Mars lander to land on Mars required fuel and a parachute.
Gravity assisted acceleration and de-acceleration is accomplished by exchanging angular momentum with a larger rotating object and is based on conservation of angular momentum.