  A new fine grid of nonlinear convective pulsation models for the so-called
"bump Cepheids" is presented to investigate the Hertzprung progression (HP)
phenomenon shown by their light and radial pulsation velocity curves. The
period corresponding to the center of the HP is investigated as a function of
various model assumptions, such as the efficiency of super-adiabatic
convection, the mass-luminosity relation, and the metal and helium abundances.
The assumed mass-luminosity relation is found to significantly affect the
phenomenon but variations in the chemical composition as well as in the stellar
mass (at fixed mass-luminosity relation) also play a key role in determining
the value of the HP center period. Finally, the predictive capability of the
presented theoretical scenario is tested against observed light curves of bump
Cepheids in the ESA Gaia database, also considering the variation of the
pulsation amplitudes and of the Fourier parameters $R_{21}$ and $\Phi_{21}$
with the pulsation period. A qualitative agreement between theory and
observations is found for what concerns the evolution of the light curve
morphology as the period moves across the HP center, as well for the pattern in
period-amplitude, period-$R21$ and period-$\Phi_{21}$ planes. A larger sample
of observed Cepheids with accurate light curves and metallicities is required
in order to derive more quantitative conclusions.
