We present new gas kinematic observations with the OSIRIS instrument at the GTC for galaxies in the Cl1604 cluster system at z=0.9. These observations together with a collection of other cluster samples at different epochs analyzed by our group are used to study the evolution of the Tully-Fisher, velocity-size and stellar mass-angular momentum relations in dense environments over cosmic time. We use 2D and 3D spectroscopy to analyze the kinematics of our cluster galaxies and extract their maximum rotation velocities (Vmax). Our methods are consistently applied to all our cluster samples which make them ideal for an evolutionary comparison. Up to redshift one, our cluster samples show evolutionary trends compatible with previous observational results in the field and in accordance with semianalytical models and hydrodynamical simulations concerning the Tully-Fisher and velocity-size relations. However, we find a factor 3 drop in disk sizes and an average B-band luminosity enhancement of 2 mag by z=1.5. We discuss the role that different cluster-specific interactions may play in producing this observational result. In addition, we find that our intermediate-to-high redshift cluster galaxies follow parallel sequences with respect to the local specific angular momentum-stellar mass relation, although displaying lower angular momentum values in comparison with field samples at similar redshifts. This can be understood by the stronger interacting nature of dense environments with respect to the field.