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Context. The fragmentation mode of high-mass molecular clumps and the accretion processes that form the most massive stars (M ≳ 8 M⊙) are still not well understood. A growing number of case studies have found massive young stellar objects (MYSOs) to harbour disk-like structures, painting a picture that the formation of high-mass stars may proceed through disk accretion, similar to that of lower-mass stars. However, the properties of such structures have yet to be uniformly and systematically characterised. Aims. The aim of this work is to uniformly study the kinematic properties of a large sample of MYSOs and characterise the stability of possible circumstellar disks against gravitational fragmentation. Methods. We have undertaken a large observational programme (CORE) making use of interferometric observations from the Northern Extended Millimetre Array (NOEMA) for a sample of 20 luminous (L > 104L⊙) protostellar objects in the 1.37 mm wavelength regime in both continuum and spectral line emission, reaching 0.4″ resolution (800 au at 2 kpc). Results. We present the gas kinematics of the full sample and detect dense gas emission surrounding 15 regions within the CORE sample. Using the dense gas tracer CH3CN, we find velocity gradients across 13 cores perpendicular to the directions of bipolar molecular outflows, making them excellent disk candidates. The extent of the CH3CN emission tracing the disk candidates varies from 1800 to 8500 au. Analysing the free-fall to rotational timescales, we find that the sources are rotationally supported. The rotation profiles of some disk candidates are well described by differential rotation while for others the profiles are poorly resolved. Fitting the velocity profiles with a Keplerian model, we find protostellar masses in the range of ~ 10–25 M⊙. Modelling the level population of CH3CN (12K–11K) K = 0–6 lines, we present temperature maps and find median temperature in the range 70–210 K with a diversity in distributions. Radial profiles of the specific angular momentum (j) for the best disk candidates span a range of 1–2 orders of magnitude, on average ~10−3 km s−1 pc, and they follow j ∝ r1.7, which is consistent with a poorly resolved rotating and infalling envelope-disk model. Studying the Toomre stability of the disk candidates, we find almost all (11 out of 13) disk candidates to be prone to fragmentation due to gravitational instabilities at the scales probed by our observations, as a result of their high disk to stellar mass ratio. In particular, disks with masses greater than ~ 10–20% of the mass of their host (proto)stars are Toomre unstable, and more luminous YSOs tend to have disks that are more massive compared to their host star and hence more prone to fragmentation. Conclusions. In this work, we show that most disk structures around high-mass YSOs are prone to disk fragmentation early in their formation due to their high disk to stellar mass ratio. This impacts the accretion evolution of high-mass protostars which will have significant implications for the formation of the most massive stars. [ABSTRACT FROM AUTHOR]