How does a convergent boundary move?
The location of a convergent boundary stays reasonably constant.
When an ocean plate converges with a continental plate. The Ocean plate is driven down creating a subduction zone. The subduction zone often creates an ocean trench at the point the two plates meet.
The continental plate is pushed upward as the ocean plate is pushed under the continental plate. The longer the motion bringing the two plates together the deeper the trench and the higher the mountains.
However the point of contact on the earth's surface remains relatively static.
When two continental plates collide due to a convergent boundary both plates are buckled upwards. An example is the where the Indian sub continent collides with the Asian continent. The Indian subcontinent is slowly being destroyed as the mountains grow higher. ( now the highest in the world) The geographic location of the point of collision remains somewhat constant.
A transverse fault like the San Andres along the California coast is a place where the convergent boundary may be considered to change. The Pacific plate is colliding with the North American plate and is sliding north while the North American plate is moving south.
It is predicted that the land portion of the Pacific plate presently attached to the North American plate will fall back into the Pacific Ocean near San Francisco.