A tire layer application assembly and a method for applying a tire layer around a substantially circular circumferential surface of an object. The assembly is provided with a drive which, during application, is coupled to the object for driving the object in a driven rotation about a concentric axis of rotation with respect to the circumferential surface, wherein the assembly includes a tracking device which, during application, is arranged to be fixed to a part of the tire layer that is applied to the circumferential surface, wherein the tracking device is rotatable in a tracking rotation about the axis of rotation of the object, independent of the driven rotation of the object about the same axis of rotation.

A rubber strip is transported to a molding drum. At this stage, a cutter cuts the rubber strip into a predetermined length. If a sensor detects the finishing end of the rubber strip when the rubber strip is transported, a sucker, which transports the rubber strip while drawing the rubber strip, is temporarily stopped or the movement speed of the sucker is decreased. As a result, a top portion of the rubber strip is extended in the longitudinal direction of the rubber strip. This compensates for the amount of shrinkage of the rubber strip after extrusion, thus restraining shrinkage of the rubber strip.