If a system experiences no external force, the center-of-mass of the system will remain at rest, or will move at constant velocity if it is already moving.
If there is an external force, the center of mass accelerates according to F = ma.
Basically, the center-of-mass of a system can be treated as a point mass, following Newton's Laws.
If an object is thrown into the air, different parts of the object can follow quite complicated paths, but the center-of-mass will follow a parabola.
If an object explodes, the different pieces of the object will follow seemingly independent paths after the explosion. The center of mass, however, will keep doing what it was doing before the explosion. This is because an explosion involves only internal forces.