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; and - 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 the actuation system comprises at least one first transmission member and one second transmission member each having a first connection portion (13a, 14a) and a second connection portion (13b, 14b); wherein the first connection portion of the first transmission member includes two coaxial oscillation pins, each oscillation pin being fixed in rotation with respect to the first transmission member and being engaged in a rotation seat, defined on a respective side wall of the housing structure; wherein 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 first transmission element; wherein the first connection portion of the second transmission member is connected in an articulated way to the second connection portion of the first transmission member for turning according to a movable axis of rotation, parallel to the fixed axis of rotation, the second connection portion of the second transmission member being connected to one of the first part of chamber and the second part of chamber for causing a displacement thereof; wherein the oscillation pins have, in cross section, a non-circular geometry; characterized in that: the side walls of the housing structure have, in homologous positions, transverse passages that extend from the rotation seats, the transverse passages having in particular a dimension of width that is at least slightly greater than a dimension of width of the oscillation pins of the first transmission member; and the first transmission member is constrained to the second transmission member so as to move in an angular range that does not include an angular position of the first transmission member such that the corresponding oscillation pins can transversely slide out of the rotation seats through the aforesaid transverse passages. - 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; and - 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 side walls of the housing structure have, in homologous positions, transverse passages that extend from the rotation seats, the transverse passages having in particular a dimension of width that is at least slightly greater than a dimension of width of the oscillation pins of the first transmission member; and the first transmission member is constrained to the second transmission member so as to move in an angular range that does not include an angular position of the first transmission member such that the corresponding oscillation pins can transversely slide out of the rotation seats through the aforesaid transverse passages.