An assembly for frothing milk, including: - an air supply assembly including an air source and an air channel having an air inlet and a downstream end, the air source being connected to the air inlet; - a fluid channel extending from a fluid inlet to a fluid outlet, the fluid channel subsequently including: ~ an air inlet emanation point to which the downstream end of the air channel is connected; ~ a frothing unit including a pump; and ~ a flow-through heating unit that bounds a heating unit fluid channel that is a part of the fluid channel, wherein the flow-through heating unit has a powered state and a non-powered state, characterized in that the flow-through heating unit has a thermal mass which is so small that, when the heating unit is in the non-powered state, an amount of 40-60 ml frothed milk with a temperature of less than 7°C at the fluid inlet, remains below 20°C, even when it has passed through the flow-through heating unit fluid channel merely 10 s after the flow-through heating unit has been switched from the powered state for producing hot frothed milk to the non-powered state for producing cold frothed milk; wherein the flow-through heating unit is a thick-film-flow-through heating unit including a thick-film heating element; wherein the thick-film heating element includes a metal heat conductive plate which on a first side is coated with a di-electric coating on which a track of electrically conductive material is applied having an electrical resistance so that heat is generated by the track when an electric current flows through the track of electrically conductive material; and wherein the thick-film-flow-through heating-unit includes a second plate that is connected with a contact side thereof to a second side of the metal plate, wherein the second plate includes a channel structure that has an open side at the contact side and wherein the first plate closes off the open side of the channel structure so as to bound the heating unit fluid channel. - an air supply assembly including an air source and an air channel having an air inlet and a downstream end, the air source being connected to the air inlet; - a fluid channel extending from a fluid inlet to a fluid outlet, the fluid channel subsequently including: ~ an air inlet emanation point to which the downstream end of the air channel is connected; ~ a frothing unit including a pump; and ~ a flow-through heating unit that bounds a heating unit fluid channel that is a part of the fluid channel, wherein the flow-through heating unit has a powered state and a non-powered state, ~ an air inlet emanation point to which the downstream end of the air channel is connected; ~ a frothing unit including a pump; and ~ a flow-through heating unit that bounds a heating unit fluid channel that is a part of the fluid channel, wherein the flow-through heating unit has a powered state and a non-powered state, the flow-through heating unit has a thermal mass which is so small that, when the heating unit is in the non-powered state, an amount of 40-60 ml frothed milk with a temperature of less than 7°C at the fluid inlet, remains below 20°C, even when it has passed through the flow-through heating unit fluid channel merely 10 s after the flow-through heating unit has been switched from the powered state for producing hot frothed milk to the non-powered state for producing cold frothed milk; wherein the flow-through heating unit is a thick-film-flow-through heating unit including a thick-film heating element; wherein the thick-film heating element includes a metal heat conductive plate which on a first side is coated with a di-electric coating on which a track of electrically conductive material is applied having an electrical resistance so that heat is generated by the track when an electric current flows through the track of electrically conductive material; and wherein the thick-film-flow-through heating-unit includes a second plate that is connected with a contact side thereof to a second side of the metal plate, wherein the second plate includes a channel structure that has an open side at the contact side and wherein the first plate closes off the open side of the channel structure so as to bound the heating unit fluid channel.