A machine for the preparation of liquid products via capsules having a substantially cup-shaped body (6a-6d) that houses a dose of at least one substance that can form a liquid product via a preparation fluid, such as water and/or steam, the machine having a delivery assembly, which comprises a stationary structure and a passage for introduction of capsules, the assembly moreover comprising: - a preparation chamber that includes a first part of chamber, having a housing configured for receiving at least partially a capsule and enabling delivery of the liquid product from the preparation chamber, and a second part of chamber configured for introduction of the fluid into the capsule; - an actuation system controllable for causing movements of the second part of chamber with respect to the first part of chamber, between a spaced apart position and a close position; - a supporting arrangement (8, 26-28) comprising a movable supporting member, which is able to keep the capsule in a position that is substantially coaxial to the parts of chamber (4, 5), and a first stop element and a second stop element for the supporting member, wherein the supporting member is mounted displaceable in a corresponding guide provided in a lower part of a body of the second part of chamber, between an advanced position of retention, in which the capsule is kept in the aforesaid position that is substantially coaxial to the parts of chamber (4, 5), and a retracted position of release, in which the capsule can drop towards an outlet passage of the delivery assembly; wherein the supporting member has a body that is able to interfere mechanically with the first stop element during movement of the second part of chamber towards the close position and is able to interfere mechanically with the second stop element during movement of the second part of chamber towards the spaced apart position, in such a way that the supporting member is displaceable from the position of retention to the position of release during movement of the second part of chamber towards the close position and is displaceable from the position of release to the position of retention during movement of the second part of chamber towards the spaced apart position, the machine being characterized in that at least one of the body of the supporting member and the second stop element comprises a first elastic element and in that the supporting arrangement (8, 26-28) is configured for enabling, with the second part of chamber in the spaced apart position, a temporary recession of the supporting member towards the position of release, against the action of the first elastic element and following upon application on the capsule of a force of thrust towards the outlet passage. - a preparation chamber that includes a first part of chamber, having a housing configured for receiving at least partially a capsule and enabling delivery of the liquid product from the preparation chamber, and a second part of chamber configured for introduction of the fluid into the capsule; - an actuation system controllable for causing movements of the second part of chamber with respect to the first part of chamber, between a spaced apart position and a close position; - a supporting arrangement (8, 26-28) comprising a movable supporting member, which is able to keep the capsule in a position that is substantially coaxial to the parts of chamber (4, 5), and a first stop element and a second stop element for the supporting member,