  We explore the multiwavelength properties of AGN host galaxies for different
classes of radio-selected AGN out to z$\lesssim$6 via a multiwavelength
analysis of about 7700 radio sources in the COSMOS field. The sources were
selected with the Very Large Array (VLA) at 3 GHz (10 cm) within the VLA-COSMOS
3 GHz Large Project, and cross-matched with multiwavelength ancillary data.
This is the largest sample of high-redshift (z$\lesssim$6) radio sources with
exquisite photometric coverage and redshift measurements available. We
constructed a sample of moderate-to-high radiative luminosity AGN (HLAGN) via
spectral energy distribution (SED) decomposition combined with standard X-ray
and mid-infrared diagnostics. Within the remainder of the sample we further
identified low-to-moderate radiative luminosity AGN (MLAGN) via excess in radio
emission relative to the star formation rates in their host galaxies. We show
that AGN power in HLAGN occurs predominantly in radiative form, while MLAGN
display a substantial mechanical AGN luminosity component. We found significant
differences in the host properties of the two AGN classes, as a function of
redshift. At z$<$1.5, MLAGN appear to reside in significantly more massive and
less star-forming galaxies compared to HLAGN. At z$>$1.5, we observed a
reversal in the behaviour of the stellar mass distributions with the HLAGN
populating the higher stellar mass tail. We interpret this finding as a
possible hint of the downsizing of galaxies hosting HLAGN, with the most
massive galaxies triggering AGN activity earlier than less massive galaxies,
and then fading to MLAGN at lower redshifts. Our conclusion is that HLAGN and
MLAGN samples trace two distinct galaxy and AGN populations in a wide range of
redshifts, possibly resembling the radio AGN types often referred to as
radiative- and jet-mode (or high- and low-excitation), respectively.
