  We present new theoretical tools, based on fluctuational electrodynamics and
the integral-equation approach to computational electromagnetism, for numerical
modeling of forces and torques on bodies of complex shapes and materials due to
emission of thermal radiation out of thermal equilibrium. This extends our
recently-developed fluctuating-surface-current (FSC) and
fluctuating-volume-current (FVC) techniques for radiative heat transfer to the
computation of non-equilibrium fluctuation-induced forces and torques; as we
show, the extension is non-trivial due to the greater computational cost of
modeling radiative momentum transfer, including new singularities that must be
carefully neutralized. We introduce a new analytical cancellation technique
that addresses these challenges and allows, for the first time, accurate and
efficient prediction of non-equilibrium forces and torques on bodies of
essentially arbitrary shapes---including asymmetric and chiral particles---and
complex material properties, including continuously-varying and anisotropic
dielectrics. We validate our approach by showing that it reproduces known
results, then present new numerical predictions of non-equilibrium
self-propulsion, self-rotation, and momentum-transfer phenomena in complex
geometries that would be difficult or impossible to study with existing
methods. Our findings indicate that the fluctuation-induced dynamics of
micron-size room-temperature bodies in cold environments involve microscopic
length scales but macroscopic time scales, with typical linear and angular
velocities on the order of microns/second and radians/second; For a
micron-scale gear driven by thermal radiation from a nearby chiral emitter, we
find a strong and non-monotonic dependence of the magnitude and even the
\textit{sign} of the induced torque on the temperature of the emitter.
