  We examine the gravitational collapse and black hole formation of multiple
non--spherical configurations constructed from Szekeres dust models with
positive spatial curvature that smoothly match to a Schwarzschild exterior.
These configurations are made of an almost spherical central core region
surrounded by a network of "pancake-like" overdensities and voids with spatial
positions prescribed through standard initial conditions. We show that a full
collapse into a focusing singularity, without shell crossings appearing before
the formation of an apparent horizon, is not possible unless the full
configuration becomes exactly or almost spherical. Seeking for black hole
formation, we demand that shell crossings are covered by the apparent horizon.
This requires very special fine-tuned initial conditions that impose very
strong and unrealistic constraints on the total black hole mass and full
collapse time. As a consequence, non-spherical non-rotating dust sources cannot
furnish even minimally realistic toy models of black hole formation at
astrophysical scales: demanding realistic collapse time scales yields huge
unrealistic black hole masses, while simulations of typical astrophysical black
hole masses collapse in unrealistically small times. We note, however, that the
resulting time--mass constraint is compatible with early Universe models of
primordial black hole formation, suitable in early dust-like environments.
Finally, we argue that the shell crossings appearing when non-spherical dust
structures collapse are an indicator that such structures do not form galactic
mass black holes but virialise into stable stationary objects.
