This paper concludes our semi-analytic study of preheating in inflationary
models comprised of multiple scalar fields coupled nonminimally to gravity.
Using the covariant framework of paper I in this series, we extend the
rigid-spacetime results of paper II by considering both the expansion of the
universe during preheating, as well as the effect of the coupled metric
perturbations on particle production. The adiabatic and isocurvature
perturbations are governed by different effective masses that scale differently
with the nonminimal couplings and evolve differently in time. The effective
mass for the adiabatic modes is dominated by contributions from the coupled
metric perturbations immediately after inflation. The metric perturbations
contribute an oscillating tachyonic term that enhances an early period of
significant particle production for the adiabatic modes, which ceases on a
time-scale governed by the nonminimal couplings $\xi_I$. The effective mass of
the isocurvature perturbations, on the other hand, is dominated by
contributions from the fields' potential and from the curvature of the
field-space manifold (in the Einstein frame), the balance between which shifts
on a time-scale governed by $\xi_I$. As in papers I and II, we identify
distinct behavior depending on whether the nonminimal couplings are small
($\xi_I \lesssim {\cal O} (1)$), intermediate ($\xi_I \sim {\cal O} (1 - 10)$),
or large ($\xi_I \geq 100$).
