Using data from SDO, PROBA2 and other spacecraft, Fainshtein and Egorov (2013) have discovered processes accompanying initiation of six limb CMEs and have studied features of their motion. The said CMEs occurred after eruption of prominence or hot emission loop and were associated with X-ray flares. The follow-up study of the CMEs, associated with the prominence eruption, showed that the formation of such mass ejections and the initial stage of their motion may be characterised by special features. In this work, we give examples of CMEs with such features. We have revealed a positive correlation between the height of the CME-related eruptive prominence and the height of the frontal structure of CMEs measured before they began to move. By analysing two of the CMEs, using SDO data, we found out that the kinematics of CME body and its related shock differs considerably. We have established that the time dependence of shock position and velocity obtained from SDO data is in agreement with theoretical dependencies of variation in these motion parameters with time in the context of self-similar motion of an explosion shock. We have concluded that the shock are not piston-like with the CME body acting as a piston.