If you are after a realistic behavior, I would use physics for that. Your boxes should be simulating physics and your conveyor belt should be applying a force to all the boxes that are in contact with it. Those that are not but simply lay on top of others that are will be dragged along by the physical contacts with those under/around them.
You can build the conveyor belt out of static meshes with collisions that you can move kinematically (AddActorWorldOffset) or you can just animate it and define a region on top of it where its drag force is applied to any cube which overlaps with it.
This may be a useful reference to get you close to what you are after: Layout