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We explore the multiwavelength properties of AGN host galaxies for different classes of radio-selected AGN out to z ≲ 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 ≲ 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 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 at each redshift our HLAGN havesystematically higher radiative luminosities than MLAGN and that their AGN power 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, whose properties might depend on the different availability of cold gas supplies.