Context. The Vela-Puppis region is known to host the Vela OB2 association as well as several young clusters featuring OB and pre-main-sequence stars. Several spatial and kinematic subgroups have been identified in recent years. Aims. By grouping stars based on their positions and velocity, we can address the question of the dynamical history of the region and the mechanisms that drove stellar formation. The Gaia DR2 astrometry and photometry enables us to characterise the 3D spatial and 3D kinematic distribution of young stars and to estimate the ages of the identified components. Methods. We used an unsupervised classification method to group stars based on their proper motions and parallax. We studied the expansion rates of the different identified groups based on 3D velocities and on corrected tangential velocities. We used theoretical isochrones to estimate ages. Results. The young stars can be separated into seven main groups of different ages and kinematical distribution. All groups are found to be expanding, although the expansion is mostly not isotropic. Conclusions. The size of the region, the age substructure, and the anisotropic expansion rates are compatible with a prolonged period of star formation in a turbulent molecular cloud. The current kinematics of the stars cannot be explained by internal processes alone (such as gas expulsion).