  The growing amount of integral-field spectroscopic data creates an increased
demand for understanding kinematic peculiarities that carry valuable
information about the evolution of the host galaxies. For kinematically
distinct cores (KDCs), a number of formation mechanisms have been proposed, but
it is still unclear which of them commonly occur in the Universe. We aim to
address the KDC formation in the cosmological context. We used the publicly
available data of the large-scale hydrodynamic cosmological simulation
Illustris. We identify 134 KDCs, study their properties, and follow their
evolution back in time. Examples of four galaxies hosting KDCs are presented
and described in detail. The masses of the KDC hosts follow the general
distribution of the Illustris galaxies, with a possible slight preference
towards massive galaxies. KDCs can be long-lived features, with their formation
epochs roughly uniformly distributed in look-back times 0-11.4 Gyr, and they
can survive even major or multiple subsequent mergers. There is no single
channel of KDC formation, but mergers seem to be the formation mechanism for
about 60% of KDCs with a significant preference for major mergers and with the
percentage being higher among massive hosts. Other KDCs formed during a
pericentric passage or flyby of another galaxy, by precession of a previously
formed rapidly rotating core, or without an obvious external cause. The mean
mass-weighted stellar age inside the KDC radius is either about the same as the
look-back time of the KDC formation or older. Although the radii of our KDCs
are on average larger than observed, we find that younger stellar ages are
typically associated with smaller KDCs. A significant fraction of KDC hosts
possess stellar shells formed during mergers that led to KDCs within the last 5
Gyr, or double peaks in their velocity dispersion maps.
