  Context: Open clusters' dynamical evolution is driven by stellar evolution,
internal dynamics and external forces, which according to dynamical
simulations, will evaporate them in a timescale of about 1 Ga. However, about
10\% of the known open clusters are older. They are special systems whose
detailed properties are related to their dynamical evolution and the balance
between mechanisms of cluster formation and dissolution. Aims:We investigate
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 identify members using Gaia EDR3 data up to a distance of 150 pc
from each cluster's centre. We investigate 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 with PYTHON Limepy and spherical
potential escapers stitched SPES. Results: All the studied clusters seem more
extended than previously reported in the literature. The spatial distributions
of three of them show some structures aligned with their orbits. They may be
related to the existence of extra tidal stars. In fact, we find that about 20 %
of their members have enough 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 found clear evidences
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 the one observed among globular clusters.
