  Solar system, exoplanet and stellar science rely on transits, eclipses and
occultations for dynamical and physical insight. Often, the geometry of these
configurations are modelled by assuming a particular viewpoint. Here, instead,
I derive user-friendly formulae from first principles independent of viewpoint
and in three dimensions. I generalise the results of Veras & Breedt (2017) by
(i) characterising three-body systems which are in transit but are not
necessarily perfectly aligned, and by (ii) incorporating motion. For a given
snapshot in time, I derive explicit criteria to determine whether a system is
in or out of transit, if an eclipse is total or annular, and expressions for
the size of the shadow, including their extreme values and a condition for
engulfment. These results are exact. For orbital motion, I instead obtain
approximate results. By assuming fixed orbits, I derive a single implicit
algebraic relation which can be solved to obtain the frequency and duration of
transit events -- including ingresses and egresses -- for combinations of
moons, planets and stars on arbitrarily inclined circular orbits; the eccentric
case requires the solution of Kepler's equation but remains algebraic. I prove
that a transit shadow -- whether umbral, antumbral or penumbral -- takes the
shape of a parabolic cylinder, and finally present geometric constraints on
Earth-based observers hoping to detect a three-body syzygy (or perfect
alignment) -- either in extrasolar systems or within the solar system --
potentially as a double annular eclipse.
