An apparatus (100), and relative method, for producing a semi-finished product (50) comprising a strip of sealing compound (51) for building tyres (1), comprising at least one unsaturated styrene thermoplastic elastomer, one bonding resin and one cross-linking agent; wherein the apparatus comprises a continuous mixer (101 ) comprising an inner chamber (103) for receiving the components of the sealing compound and to continuously mix them until the sealing compound is obtained; a buffer device (110) having a respective inner chamber (111) and coupled in series with structural continuity to the continuous mixer, downstream of the latter; a gear pump (120) having a respective inner chamber (121) and a gear system (122) housed in the respective inner chamber, and coupled with structural continuity to the buffer device downstream of the latter, an extrusion head (130) coupled directly and with structural continuity to the gear pump, downstream of the latter, in a manner such to receive in input the sealing compound from the gear pump and deliver in output the sealing compound in the form of a strip having predetermined transverse size, wherein the buffer device (110) receives the sealing compound from the continuous mixer (101) and feeds the gear pump.