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.
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