On a building drum, anchoring structures to beads, carcass plies, a belt structure and/or other annular composite components are sequentially assembled with a liner, sidewalls, sidewall inserts, under-belt inserts or other annular elastomer components. Each of the sidewall inserts, sidewalls, or under-belt inserts is made as a one piece construction in a service station operating far away from the building drum for storage on a movable storing device. Actuating members operating on the movable storing device carry the annular elastomer components from the service station to the building drum. A transfer member picks up the annular elastomer components from the movable storing device to transfer them to the building drum.