  (Abridged) Observational surveys of entire star-forming regions have provided
evidence of power-law correlations between the disc properties and the stellar
mass, especially the disc mass (${M_d \propto M_*}^{\lambda_m}$) and the
accretion rate ($\dot M \propto {M_*}^{\lambda_{acc}}$). Whether the secular
disc evolution affects said correlations is still debated: while the purely
viscous scenario has been probed, other mechanisms could impact differently. We
study the evolution of the slopes $\lambda_m$ and $\lambda_{acc}$ in the
wind-driven and hybrid case and compare it to the viscous prediction, using a
combination of analytical calculations and numerical simulations (performed
with the 1D population synthesis code Diskpop, that we also present and
release). Assuming $M_d(0) \propto {M_*}^{\lambda_{m, 0}}$ and $\dot M(0)
\propto {M_*}^{\lambda_{acc, 0}}$ as initial conditions, we find that viscous
and hybrid accretion preserve the shape of the correlations and evolve their
slope; on the other hand, MHD winds change the shape of the correlations,
bending them according to the scaling of the accretion timescale with the
stellar mass. We also show how a spread in the initial conditions conceals this
behaviour. We then analyse the impact of disc dispersal, and find that the
currently available sample sizes ($\sim 30$ discs at 5 Myr) introduce
stochastic oscillations in the slopes evolution, which dominate over the
physical signatures. Increasing the sample size could mitigate this issue:
$\sim 140$ discs at 5 Myr, corresponding to the complete Upper Sco sample,
would give small enough error bars to use the evolution of the slopes as a
proxy for the driving mechanism of disc evolution. Finally, we discuss how the
observational claim of steepening slopes necessarily leads to an initially
steeper $M_d - M_*$ correlation with respect to $\dot M - M_*$.
