  Through second order in perturbative general relativity, a small compact
object in an external vacuum spacetime obeys a generalized equivalence
principle: although it is accelerated with respect to the external background
geometry, it is in free fall with respect to a certain \emph{effective} vacuum
geometry. However, this single principle takes very different mathematical
forms, with very different behaviors, depending on how one treats perturbed
motion. Furthermore, any description of perturbed motion can be altered by a
gauge transformation. In this paper, I clarify the relationship between two
treatments of perturbed motion and the gauge freedom in each. I first show
explicitly how one common treatment, called the Gralla-Wald approximation, can
be derived from a second, called the self-consistent approximation. I next
present a general treatment of smooth gauge transformations in both
approximations, in which I emphasise that the approximations' governing
equations can be formulated in an invariant manner. All of these analyses are
carried through second perturbative order, but the methods are general enough
to go to any order. Furthermore, the tools I develop, and many of the results,
should have broad applicability to any description of perturbed motion,
including osculating-geodesic and two-timescale descriptions.
