  The standard model of cosmology is based on homogeneous-isotropic solutions
of Einstein's equations. These solutions are known to be gravitationally
unstable to local inhomogeneous perturbations, commonly described as evolving
on a background given by the same solutions. In this picture, the FLRW
backgrounds are taken to describe the average over inhomogeneous perturbations
for all times. We study in the present article the (in)stability of FLRW dust
backgrounds within a class of averaged inhomogeneous cosmologies. We examine
the phase portraits of the latter, discuss their fixed points and orbital
structure and provide detailed illustrations. We show that FLRW cosmologies are
unstable in some relevant cases: averaged models are driven away from them
through structure formation and accelerated expansion. We find support for the
proposal that the dark components of the FLRW framework may be associated to
these instability sectors. Our conclusion is that FLRW cosmologies have to be
considered critically as for their role to serve as reliable models for the
physical background.
