Does costochondritis typically move around the chest from left to center of the chest?

Sometimes. Costochondritis is an inflammation in the small joints between the sternum-breast bone - and the ribs. Costchondritis is often idiopathic - meaning we don't know what causes it, but is frequently assumed to be viral in origin. At other times it is directly related to exercise using the upper extremities or trauma to the chest. It can move around the chest, but discuss this with your doctor.
It could. Costochondritis can affect any of the cartilages that connect the ribs to the sternum (or breast bone), so can affect either side of the breast bone or along either costal margin (the bottom edges of the rib cage).