A machine (1) for the preparation of liquid products via capsules comprises a delivery assembly (2) having an actuation system that can be controlled to cause relative displacements between a first part and a second part of a preparation chamber. The actuation system comprises at least one first transmission member (13) and one second transmission member (14a). The first transmission member (13) includes two pins (13a), which are coaxial and are engaged in corresponding seats (3d), defined on re-spective side walls of the housing structure (3). The seats (3d) define a fixed axis of rotation of the first transmission element (13). The second transmission member (14a) is connected in an articulated way to the first transmission member (13) for turning about a movable axis of rotation, parallel to the fixed axis of rotation, and to one of the first part of chamber and the second part of chamber. The pins (13a) have, in cross section, a non-circular geometry, and the side walls of the housing structure (3) have, in homologous positions, transverse passages (3f) that extend from the seats (3d). The first transmission member (13) is constrained to the second transmission member (14a) so as to move in an angular range that does not include an angular position of the first transmission member (13) such that the corresponding pins (13a) would slide out transversely of the rotation seats (3d) through the aforesaid transverse passages (3f).