Context.The dynamical evolution of open clusters is driven by stellar evolution, internal dynamics, and external forces, which according to dynamical simulations will lead to their evaporation over a timescale of about 1 Ga. However, about 10% of the known open clusters are older. These latter are special systems whose detailed properties are related to their dynamical evolution and the balance between mechanisms of cluster formation and dissolution.Aims.We investigated the spatial distribution and structural parameters of six open clusters older than 1 Ga in order to constrain their dynamical evolution and longevity.Methods.We identified members usingGaiaEDR3 data up to a distance of 150 pc from the centre of each cluster. We investigated the spatial distribution of stars inside each cluster to understand their degree of mass segregation. Finally, in order to interpret the obtained radial density profiles, we reproduced them using the lowered isothermal model explorer withPYthon(LIMEPY) and the spherical potential escapers stitched (SPES) models.Results.All the studied clusters appear to be more extended than previously reported in the literature. The spatial distributions of three of them show some structures aligned with their orbits. These structures may be related to the existence of extra tidal stars. Moreover, we find that about 20% of their members have sufficient energy to leave the systems or are already unbound. Together with their initial masses, their distances to the Galactic plane may play significant roles in their survival. We find clear evidence that the most dynamically evolved clusters do not fill their Roche volumes, appearing more concentrated than the others. Finally, we find a cusp–core dichotomy in the central regions of the studied clusters, which shows some similarities to that observed among globular clusters.