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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. Massive disks are prone to fragmentation via gravitational instabilities due to high gas densities and accretion rates. Therefore, it is important to study the stability of such disks in order to put into context the role of disk fragmentation in setting the final stellar mass distribution in high-mass star forming regions. 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 program (CORE) making use of interferometric observations from the Northern Extended Millimetre Array (NOEMA) for a sample of 20 luminous (L > 104 L) protostellar objects in the 1.37 mm wavelength regime in both continuum and spectral line emission, reaching 0.400 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 − 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