A machine for the preparation of liquid products via capsules, comprising a delivery assembly having: - a housing structure; - a preparation chamber, which comprises a first part and a second part, the first part of chamber having a housing configured for receiving at least partially one said capsule and enabling delivery of the liquid product from the preparation chamber, the second part of chamber being configured for introducing a preparation fluid into said capsule, in particular, water and/or steam under pressure; - an actuation system controllable for causing relative displacements between the first part of chamber and the second part of chamber, between a spaced apart position and a close position; wherein defined in homologous positions on the side walls of the housing structure are respective guides with arc-shaped development; the machine being characterized in that: - the actuation system comprises at least one rocker and one connecting-rod member, each having a first connection portion (13a, 14a) and a second connection portion (13b, 14b); - the first connection portion of the rocker includes two coaxial oscillation pins, each oscillation pin being fixed in rotation with respect to the rocker and being engaged in a rotation seat, defined on a respective side wall of the housing structure; - the rotation seats have a surface of rotation with which an external profile of the respective oscillation pin co-operates for defining a fixed axis of rotation of the rocker; - the second connection portion and the first connection portion of the rocker and of the connecting-rod member, respectively, are connected in an articulated way by means of a first connection pin for turning according to a movable axis of rotation, parallel to the fixed axis of rotation; - the two opposite ends of the first connection pin are engaged in said guides with arc-shaped development defined in homologous positions on the side walls of the housing structure, the guides with arc-shaped development determining an angular range of oscillation of the rocker; - the second connection portion of the connecting-rod member is connected in an articulated way to one of the first part of chamber and the second part of chamber, the latter being constrained to a linear reciprocating movement within the housing structure; - in the position where the first and second parts of chamber (4, 5) are close to one another, the ends of the first connection pin are at a bottom of the corresponding guides with arc-shaped development; - the first connection pin is displaceable from above downwards, or else from beneath upwards, respectively, to enable passage of the first and second parts of chamber (4, 5) into the respective position where they are set close to one another, and in said position where they are close to one another the movable axis of rotation being at a lower height, or else at a greater height, respectively, than the fixed axis of rotation. - a housing structure; - a preparation chamber, which comprises a first part and a second part, the first part of chamber having a housing configured for receiving at least partially one said capsule and enabling delivery of the liquid product from the preparation chamber, the second part of chamber being configured for introducing a preparation fluid into said capsule, in particular, water and/or steam under pressure; - an actuation system controllable for causing relative displacements between the first part of chamber and the second part of chamber, between a spaced apart position and a close position; - the actuation system comprises at least one rocker and one connecting-rod member, each having a first connection portion (13a, 14a) and a second connection portion (13b, 14b); - the first connection portion of the rocker includes two coaxial oscillation pins, each oscillation pin being fixed in rotation with respect to the rocker and being engaged in a rotation seat, defined on a respective side wall of the housing structure; - the rotation seats have a surface of rotation with which an external profile of the respective oscillation pin co-operates for defining a fixed axis of rotation of the rocker; - the second connection portion and the first connection portion of the rocker and of the connecting-rod member, respectively, are connected in an articulated way by means of a first connection pin for turning according to a movable axis of rotation, parallel to the fixed axis of rotation; - the two opposite ends of the first connection pin are engaged in said guides with arc-shaped development defined in homologous positions on the side walls of the housing structure, the guides with arc-shaped development determining an angular range of oscillation of the rocker; - the second connection portion of the connecting-rod member is connected in an articulated way to one of the first part of chamber and the second part of chamber, the latter being constrained to a linear reciprocating movement within the housing structure; - in the position where the first and second parts of chamber (4, 5) are close to one another, the ends of the first connection pin are at a bottom of the corresponding guides with arc-shaped development; - the first connection pin is displaceable from above downwards, or else from beneath upwards, respectively, to enable passage of the first and second parts of chamber (4, 5) into the respective position where they are set close to one another, and in said position where they are close to one another the movable axis of rotation being at a lower height, or else at a greater height, respectively, than the fixed axis of rotation.