Process for the production of functionalized thermoplastic elastomers based on graft substrates, selected among - olefin block copolymers of the composition 80 to 98% by mass of ethylene units/2 to 20% by mass of C 3 -C 12 -olefin units or - among semicrystalline propylene/ethylene copolymers and/or C 4 - to C 12 -olefin copolymers and/or C 4 - to C 12 -diene copolymers in each case of the composition 50 to 98% by mass of propylene units/2 to 50% by mass of C 2 - and/or C 4 - to C 12 -olefin units and/or C 4 - to C 12 -diene units or - among crosslinked styrene/olefin/styrene block copolymers or styrene/olefin block copolymers, in which the following are added in a fluid mixing reactor, i.e. in a temperature-controllable mixer for fine-grained substances which ensures a freeflowing powder bed, for each 100 parts by mass of particulate graft substrate: - 0.1 to 15 parts by mass of at least one functional monomer from the group of the α,β-ethylenically unsaturated compounds comprising functional groups or 0.1 to 15 parts by mass of a monomer mixture comprising at least one of these functional monomers, and also - 0.01 to 10 parts by mass of at least one free-radical-generating initiator with a 1-hour-half-life temperature (T HL/1h) between 50 and 200°C are added and are polymerized at reaction temperatures between 40°C and the melting or softening point of the graft substrate over a reaction time between 10 and 200 min in solid-fluid phase, where this solid-phase functionalization produces graft product which comprises a grafted functional monomer and which is used as starting component for further processing, characterized in that during further processing of the graft product - 100 parts by mass of solid-phase graft product into which between 0.1 and 60 parts by mass of at least one functional monomer or of a monomer mixture comprising at least one functional monomer, and also between 0.01 and 20 parts by mass of at least one free-radical-generating initiator with a 1-hour-half-life temperature (T HL/1h) between 80 and 240°C have been mixed are continuously introduced by way of metering devices, together with 100 to 4000 parts by mass of an unmodified olefinic elastomer, into the intake region of an extruder, - the reactive extrusion is carried out at temperatures above the melting or softening point of the olefinic elastomer and - a functionalized elastomer is continuously discharged at the end of the reactor. - olefin block copolymers of the composition 80 to 98% by mass of ethylene units/2 to 20% by mass of C 3 -C 12 -olefin units or - among semicrystalline propylene/ethylene copolymers and/or C 4 - to C 12 -olefin copolymers and/or C 4 - to C 12 -diene copolymers in each case of the composition 50 to 98% by mass of propylene units/2 to 50% by mass of C 2 - and/or C 4 - to C 12 -olefin units and/or C 4 - to C 12 -diene units or - among crosslinked styrene/olefin/styrene block copolymers or styrene/olefin block copolymers, - 0.1 to 15 parts by mass of at least one functional monomer from the group of the α,β-ethylenically unsaturated compounds comprising functional groups or 0.1 to 15 parts by mass of a monomer mixture comprising at least one of these functional monomers, and also - 0.01 to 10 parts by mass of at least one free-radical-generating initiator with a 1-hour-half-life temperature (T HL/1h) between 50 and 200°C are added and are polymerized at reaction temperatures between 40°C and the melting or softening point of the graft substrate over a reaction time between 10 and 200 min in solid-fluid phase, where this solid-phase functionalization produces graft product which comprises a grafted functional monomer and which is used as starting component for further processing, characterized in that during further processing of the graft product - 100 parts by mass of solid-phase graft product into which between 0.1 and 60 parts by mass of at least one functional monomer or of a monomer mixture comprising at least one functional monomer, and also between 0.01 and 20 parts by mass of at least one free-radical-generating initiator with a 1-hour-half-life temperature (T HL/1h) between 80 and 240°C have been mixed are continuously introduced by way of metering devices, together with 100 to 4000 parts by mass of an unmodified olefinic elastomer, into the intake region of an extruder, - the reactive extrusion is carried out at temperatures above the melting or softening point of the olefinic elastomer and - a functionalized elastomer is continuously discharged at the end of the reactor.