Grafting from the diblock copolymers alpha b beta leads to block-graft copolymers of the form alpha b beta g gamma, the chemistry and architecture of which can be varied over wide limits. A family of compatibilizers for polymer blends can be prepared from one anionically produced and thus expensive copolymer, by a radically initiated grafting process, which is a comparatively cheap method. When alpha b beta is asymmetric, with the block b beta carrying the graft g gamma being short, the resulting copolymer alpha b beta/g gamma is essentially a diblock copolymer alpha b gamma. The preparation of the block-graft copolymers SbBgX, where X is methyl methacrylate (MMA) or cyclohexyl methacrylate (CHMA) is discussed. These are based on a diblock copolymer SbB of styrene and butadiene and are grafted with MMA or CMMA. These copolymers compatibilize blends of polystyrene (PS) (or poly(phenylene ether) and poly(vinyl chloride) (PVC) (or styrene-acrylonitile (SAN) copolymer). Attention is focused on the reaction kinetics as monitored by conventional methods and by transmission electron microscopy. The microscopic analysis shows the ongoing progress of the grafting reaction as well as the mutual miscibility of all polymers in the system.