  The Universe on scales $10-100~h^{-1}$ Mpc is dominated by a cosmic web of
voids, filaments, sheets and knots of galaxy clusters. These structures
participate differently in the global expansion of the Universe: from
non-expanding clusters to the above average expansion rate of voids. In this
paper we characterize Hubble expansion anisotropies in the COMPOSITE sample of
4534 galaxies and clusters. We concentrate on the dipole and quadrupole in the
rest frame of the Local Group. These both have statistically significant
amplitudes. These anisotropies, and their redshift dependence, cannot be
explained solely by a boost of the Local Group in the
Friedmann-Lema\^itre-Robertson-Walker (FLRW) model which expands isotropically
in the rest frame of the cosmic microwave background (CMB) radiation. We
simulate the local expansion of the Universe with inhomogeneous Szekeres
models, which match the standard FLRW model on $> 100~h^{-1}$ Mpc scales but
exhibit nonkinematic relativistic differential expansion on small scales. We
restrict models to be consistent with observed CMB temperature anisotropies,
while simultaneously fitting the redshift variation of the Hubble expansion
dipole. We include features to account for both the Local Void and the "Great
Attractor". While this naturally accounts for the Hubble expansion and CMB
dipoles, the simulated quadrupoles are smaller than observed. Further
refinement to incorporate additional structures may improve this. This would
enable a test of the hypothesis that some large angle CMB anomalies result from
failing to treat the relativistic differential expansion of the background
geometry, a natural feature of solutions to Einstein's equations not included
in the current standard model of cosmology.
