  Mapping the average AGN luminosity across galaxy populations and over time
encapsulates important clues on the interplay between supermassive black hole
(SMBH) and galaxy growth. This paper presents the demography, mean power and
cosmic evolution of radio AGN across star-forming galaxies (SFGs) of different
stellar masses (${M_{*}}$). We exploit deep VLA-COSMOS 3 GHz data to build the
rest-frame 1.4 GHz AGN luminosity functions at 0.1$\leq$$z$$\leq$4.5 hosted in
SFGs. Splitting the AGN luminosity function into different ${M_{*}}$ bins
reveals that, at all redshifts, radio AGN are both more frequent and more
luminous in higher ${M_*}$ than in lower ${M_*}$ galaxies. The cumulative
kinetic luminosity density exerted by radio AGN in SFGs peaks at $z$$\sim$2,
and it is mostly driven by galaxies with
10.5$\leq$$\log$(${M_{*}}$/${M_{\odot}}$)$<$11. Averaging the cumulative radio
AGN activity across all SFGs at each (${M_{*}}$,$z$) results in a "radio-AGN
main sequence" that links the time-averaged radio-AGN power
$\langle$$L_{1.4}^{{AGN}}$$\rangle$ and galaxy stellar mass, in the form:
$\log$$\langle$[$L_{1.4}^{{AGN}}$/ W Hz$^{-1}]\rangle$ = (20.97$\pm$0.16) +
(2.51$\pm$0.34)$\cdot$$\log$(1+$z$) +
(1.41$\pm$0.09)$\cdot$($\log$[${M_{*}}$/${M_{\odot}}$] -10). The super-linear
dependence on ${M_{*}}$, at fixed redshift, suggests enhanced radio-AGN
activity in more massive SFGs, as compared to star formation. We ascribe this
enhancement to both a higher radio AGN duty cycle and a brighter radio-AGN
phase in more massive SFGs. A remarkably consistent ${M_{*}}$ dependence is
seen for the evolving X-ray AGN population in SFGs. This similarity is
interpreted as possibly driven by secular cold gas accretion fueling both radio
and X-ray AGN activity in a similar fashion over the galaxy's lifetime.
