  We study the moduli space of SU(3) structure manifolds X that form the
internal compact spaces in four-dimensional N=1/2 domain wall solutions of
heterotic supergravity with flux. Together with the direction perpendicular to
the four-dimensional domain wall, X forms a non-compact 7-manifold Y with
torsionful G2 structure. We use this G2 embedding to explore how X(t) varies
along paths C(t) in the SU(3) structure moduli space. Our analysis includes the
Bianchi identities which strongly constrain the flow. We show that requiring
that the SU(3) structure torsion is preserved along the path leads to
constraints on the G2 torsion and the embedding of X in Y. Furthermore, we
study flows along which the torsion classes of X go from zero to non-zero
values. In particular, we present evidence that the flow of half-flat SU(3)
structures may contain Calabi-Yau loci, in the presence of non-vanishing
H-flux.
