  We analyze transitions between heterotic vacua with distinct gauge bundles
using two complementary methods - the effective four-dimensional field theory
and the corresponding geometry. From the viewpoint of effective field theory,
such transitions occur between flat directions of the potential energy
associated with heterotic stability walls. Geometrically, this branch structure
corresponds to smooth deformations of the gauge bundle coupled to the chamber
structure of K\"ahler moduli space. We demonstrate how such transitions can
change important properties of the effective theory, including the gauge
symmetry and the massless spectrum. Geometrically, this study is divided into
deformations of the vector bundle which preserve the rank of the gauge bundle
and those which change the rank. In the latter case, our results provide
explicit solutions to a class of Li-Yau type deformation problems. Finally, we
use the framework of stability walls and their effective theory to study
Donaldson-Thomas invariants on Calabi-Yau threefolds.
