  (Abridged) We present an investigation of kinematical imprints of AGN
feedback on the Warm Ionized gas Medium (WIM) of massive early-type galaxies
(ETGs). To this end, we take a two-fold approach that involves a comparative
analysis of Halpha velocity fields in 123 local ETGs from the CALIFA integral
field spectroscopy survey with 20 simulated galaxies from high-resolution
hydrodynamic cosmological SPHgal simulations. The latter were re-simulated for
two modeling setups, one with and another without AGN feedback. In order to
quantify the effects of AGN feedback on gas kinematics we measure three
parameters that probe deviations from simple regular rotation using the
kinemetry package. These indicators trace the possible presence of distinct
kinematic components in Fourier space (k3,5/k1), variations in the radial
profile of the kinematic major axis (sigma_PA), and offsets between the stellar
and gas velocity fields (Delta Phi). These quantities are monitored in the
simulations from a redshift 3 to 0.2 to assess the connection between black
hole accretion history, stellar mass growth and kinematical perturbation of the
WIM. Observed local massive galaxies show a broad range of irregularities,
indicating disturbed warm gas motions, irrespective of being classified via
diagnostic lines as AGN or not. Simulations of massive galaxies with AGN
feedback generally exhibit higher irregularity parameters than without AGN
feedback, more consistent with observations. Besides AGN feedback, other
processes like major merger events or infalling gas clouds can lead to elevated
irregularity parameters, but they are typically of shorter duration. More
specifically, k3,5/k1 is most sensitive to AGN feedback, whereas Delta Phi is
most strongly affected by gas infall.
