We have studied a sample of 1604 moderate-to-high radiative luminosity active galactic nuclei (HLAGN) selected at 3 GHz within the VLA-COSMOS 3 GHz Large Project. These were classified by combining multiple AGN diagnostics: X-ray data, mid-infrared data and broadband spectral energy distribution fitting. We decomposed the total radio 1.4 GHz luminosity (L1.4 GHz, TOT) into the emission originating from star formation and AGN activity by measuring the excess in L1.4 GHz, TOT relative to the infrared-radio correlation of star-forming galaxies. To quantify the excess, for each source we calculated the AGN fraction (fAGN) defined as the fractional contribution of AGN activity to L1.4 GHz, TOT. The majority of the HLAGN, (68.0 ± 1.5)%, are dominated by star-forming processes (fAGN ≤ 0.5), while (32.0 ± 1.5)% are dominated by AGN-related radio emission (0.5 <  fAGN ≤ 1). We used the AGN-related 1.4 GHz emission to derive the 1.4 GHz AGN luminosity functions of HLAGN. By assuming pure density and pure luminosity evolution models we constrained their cosmic evolution out to z ∼ 6, finding Φ*(z)∝(1 + z)(2.64 ± 0.10)+(−0.61 ± 0.04)z and L*(z)∝(1 + z)(3.97 ± 0.15)+(−0.92 ± 0.06)z. These evolutionary laws show that the number and luminosity density of HLAGN increased from higher redshifts (z ∼ 6) up to a maximum in the redshift range 1 <  z <  2.5, followed by a decline toward local values. By scaling the 1.4 GHz AGN luminosity to kinetic luminosity using the standard conversion, we estimate the kinetic luminosity density as a function of redshift. We compare our result to the semi-analytic models of radio mode feedback, and find that this feedback could have played an important role in the context of AGN-host co-evolution in HLAGN which shows evidence of AGN-related radio emission (fAGN >  0).