  Asteroid material is detected in white dwarfs (WDs) as atmospheric pollution
by metals, in the form of gas/dust discs, or in photometric transits. Within
the current paradigm, minor bodies need to be scattered, most likely by
planets, into highly eccentric orbits where the material gets disrupted by
tidal forces and then accreted onto the star. This can occur through a
planet-planet scattering process triggered by the stellar mass loss during the
post main-sequence evolution of planetary systems. So far, studies of the
$N$-body dynamics of this process have used artificial planetary system
architectures built ad hoc. In this work, we attempt to go a step further and
study the dynamical instability provided by more restrictive systems, that, at
the same time allow us an exploration of a wider parameter space: the hundreds
of multiple planetary systems found around main-sequence (MS) stars. We find
that most of our simulated systems remain stable during the MS, Red and
Asymptotic Giant Branch and for several Gyr into the WD phases of the host
star. Overall, only $\approx$ 2.3$\%$ of the simulated systems lose a planet on
the WD as a result of dynamical instability. If the instabilities take place
during the WD phase most of them result in planet ejections with just 5
planetary configurations ending as a collision of a planet with the WD. Finally
3.2$\%$ of the simulated systems experience some form of orbital scattering or
orbit crossing that could contribute to the pollution at a sustained rate if
planetesimals are present in the same system.
