Context. Open clusters (OCs) trace the evolution of the Galactic disc with great accuracy. Gaia and large ground-based spectroscopic surveys make it possible to determine their properties and study their kinematics with unprecedented precision.Aims. We study the kinematical behaviour of the OC population over time. We take advantage of the latest age determinations of OCs to investigate the correlations of the 6D phase-space coordinates and orbital properties with age. The phase-space distribution, age-velocity relation, and action distribution are compared to those of field stars. We also investigate the rotation curve of the Milky Way traced by OCs, and we compare it to that of other observational or theoretical studies.Methods. We gathered nearly 30 000 radial velocity (RV) measurements of OC members from both Gaia-RVS data and ground-based surveys and catalogues. We computed the weighted mean RV, Galactic velocities, and orbital parameters of 1382 OCs. We investigated their distributions as a function of age and by comparison to field stars.Results. We provide the largest RV catalogue available for OCs, half of it based on at least three members. Compared to field stars, we note that OCs are not on exactly the same arches in the radial-azimuthal velocity plane, while they seem to follow the same diagonal ridges in the Galactic radial distribution of azimuthal velocities. Velocity ellipsoids in different age bins all show a clear anisotropy. The heating rate of the OC population is similar to that of field stars for the radial and azimuthal components, but it is significantly lower for the vertical component. The rotation curve drawn by our sample of clusters shows several dips that match the wiggles derived from nonaxisymmetric models of the Galaxy. From the computation of orbits, we obtain a clear dependence of the maximum height and eccentricity on age. Finally, the orbital characteristics of the sample of clusters as shown by the action variables follow the distribution of field stars. The additional age information of the clusters indicates some (weak) age dependence of the known moving groups.< Réduire