We perform a comprehensive study of the stellar population properties of quiescent galaxies as a function of size and stellar mass to constrain the physical mechanism governing the stellar mass assembly and the likely evolutive scenarios that explain their growth in size. After selecting all the quiescent galaxies from the ALHAMBRA survey by the dust-corrected stellar mass$-$colour diagram, we built a shared sample of $\sim 850$ quiescent galaxies with reliable sizes from the HST. The stellar population properties were retrieved using the SED-fitting code MUFFIT with various sets of composite stellar population models. Age, formation epoch, metallicity, and extinction were studied on the stellar mass$-$size plane as function of size through a Monte Carlo approach. This accounted for uncertainties and degeneracy effects amongst stellar population properties. The stellar population properties of quiescent galaxies and their stellar mass and size since $z\sim 1$ are correlated. At fixed stellar mass, the more compact the quiescent galaxy, the older and richer in metals it is ($1$Gyr and $0.1$dex, respectively). In addition, more compact galaxies may present slight lower extinctions than their more extended counterparts at the same stellar mass ( mag). By means of studying constant regions of stellar population properties across the stellar mass$-$size plane, we obtained empirical relations to constrain the physical mechanism that governs the stellar mass assembly of the form $M\star \propto rc\alpha$, where $\alpha$ amounts to $0.50-0.55\pm 0.09$. There are indications that support the idea that the velocity dispersion is tightly correlated with the stellar content of galaxies. The mechanisms driving the evolution of stellar populations can therefore be partly linked to the dynamical properties of galaxies, along with their gravitational potential.