Exploring how radio-emitting active galactic nuclei (AGNs) behave and evolve with time is critical for understanding how AGN feedback impacts galaxy evolution. In this work, we investigate the relationship between 1.4 GHz radio continuum AGN luminosity (L^AGN_1.4), specific black hole accretion rate (s-BHAR, defined as the accretion luminosity relative to the galaxy stellar mass), and redshift, for a luminosity-complete sample of radio-selected AGNs in the VLA COSMOS 3 GHz Large Project. The sample was originally selected from radio- continuum observations at 3 GHz, and includes about 1800 radio AGNs identified via (>2 σ) radio-excess relative to the infrared-radio correlation of star-forming galaxies. We further select a subsample of over 1200 radio AGNs that is complete in (L^AGN_1.4) over different redshift ranges, out to z ∼ 4, and use X-ray stacking to calculate the average s- BHAR in each (L^AGN_1.4)-z bin. We find that the average s-BHAR is independent of (L^AGN_1.4), at all redshifts. However, we see a strong increase of s-BHAR with redshift, at fixed (L^AGN_1.4). This trend resembles the strong increase in the fraction of star-forming host galaxies [based on the (NUV − r)/(r − J) colours] with redshift, at fixed (L^AGN_1.4). A possible explanation for this similarity might imply a link between average AGN radiative power and availability of cold gas supply within the host galaxy. This study corroborates the idea that radio-selected AGNs become more radiatively efficient towards earlier epochs, independently of their radio power.