Most of the asteroids are in a more or less stable orbit around the sun. Some are in Earth-crossing orbits; however, the smaller an asteroid, the more easily it can be bumped out of its place by an exterior force. Its position and direction of movement will change. It may find itself positioned around some other asteroids, in which case it will stay in the asteroid belt.
Or, it may get affected in such a way that it is knocked toward the inner solar system, and it will begin to fall toward the sun. It may or may not be gravitationally caught by one of the inner planets before it gets to the sun. An asteroid may leave the asteroid belt if it is hit by another asteroid, which creates a force that moves it from orbit.