  We investigate the conditions under which cosmological variations in physical
`constants' and scalar fields are detectable on the surface of local
gravitationally-bound systems, such as planets, in non-spherically symmetric
background spacetimes. The method of matched asymptotic expansions is used to
deal with the large range of length scales that appear in the problem. We
derive a sufficient condition for the local time variation of the scalar fields
driving variations in 'constants' to track their large-scale cosmological
variation and show that this is consistent with our earlier conjecture derived
from the spherically symmetric problem. We perform our analysis with spacetime
backgrounds that are of Szekeres-Szafron type. They are approximately
Schwarzschild in some locality and free of gravitational waves everywhere. At
large distances, we assume that the spacetime matches smoothly onto a Friedmann
background universe. We conclude that, independent of the details of the
scalar-field theory describing the varying `constant', the condition for its
cosmological variations to be measured locally is almost always satisfied in
physically realistic situations. The very small differences expected to be
observed between different scales are quantified. This strengthens the proof
given in our previous paper that local experiments see global variations by
dropping the requirement of exact spherical symmetry. It provides a rigorous
justification for using terrestrial experiments and solar system observations
to constraint or detect any cosmological time variations in the traditional
`constants' of Nature in the case where non-spherical inhomogeneities exist.
