  This is the second in a series of papers whose aim is to generate
``adiabatic'' gravitational waveforms from the inspiral of stellar-mass compact
objects into massive black holes. In earlier work, we presented an accurate
(2+1)D finite-difference time-domain code to solve the Teukolsky equation,
which evolves curvature perturbations near rotating (Kerr) black holes. The key
new ingredient there was a simple but accurate model of the singular source
term based on a discrete representation of the Dirac-delta function and its
derivatives. Our earlier work was intended as a proof of concept, using simple
circular, equatorial geodesic orbits as a testbed. Such a source is effectively
static, in that the smaller body remains at the same coordinate radius and
orbital inclination over an orbit. (It of course moves through axial angle, but
we separate that degree of freedom from the problem. Our numerical grid has
only radial, polar, and time coordinates.) We now extend the time-domain code
so that it can accommodate dynamic sources that move on a variety of physically
interesting world lines. We validate the code with extensive comparison to
frequency-domain waveforms for cases in which the source moves along generic
(inclined and eccentric) bound geodesic orbits. We also demonstrate the ability
of the time-domain code to accommodate sources moving on interesting
non-geodesic worldlines. We do this by computing the waveform produced by a
test mass following a ``kludged'' inspiral trajectory, made of bound geodesic
segments driven toward merger by an approximate radiation loss formula.
