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 Friedmann-Lemaître-Robertson-Walker (FLRW) backgrounds are taken to describe the average over inhomogeneous perturbations for all times. We study in this paper 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 with these instability sectors. Our conclusion is that FLRW cosmologies have to be considered critically for their role to serve as reliable models for the physical background.