  The stellar content of Galactic open clusters is gradually depleted during
their evolution as a result of internal relaxation and external interactions.
The final residues of the evolution of open clusters are called open cluster
remnants, barely distinguishable from the field. We aimed to characterise and
compare the dynamical states of a set of 16 such objects. The sample also
includes 7 objects that are catalogued as dynamically evolved open clusters. We
used photometric data from the 2MASS, astrometric data from the GAIA DR2 and a
decontamination algorithm that was applied to the three-dimensional astrometric
space of proper motions and parallaxes for stars in the objects' areas. The
luminosity and mass functions and total masses for most open cluster remnants
are derived here for the first time. Our analysis used predictions of N-body
simulations to estimate the initial number of stars of the remnants from their
dissolution timescales. The investigated open cluster remnants present masses
and velocity dispersions within well-defined ranges: M between ~10-40M_Sun and
sigma_v between ~1-7km/s. Some objects in the remnant sample have a limiting
radius R_lim<~2pc, which means that they are more compact than the investigated
open clusters; other remnants have R_lim between ~2-7pc, which is comparable to
the open clusters. We suggest that the open cluster NGC2180 is entering a
remnant evolutionary stage. In general, our clusters show signals of depletion
of low-mass stars. This confirms their dynamically evolved states. We conclude
that the open cluster remnants we studied are in fact remnants of initially
very populous open clusters (No~10^3-10^4 stars). The outcome of the long-term
evolution is to bring the final residues of the open clusters to dynamical
states that are similar to each other, thus masking out the memory of the
initial formation conditions of star clusters.
