  We derive conservative, multidimensional, energy-dependent moment equations
for neutrino transport in core-collapse supernovae and related astrophysical
systems, with particular attention to the consistency of conservative
four-momentum and lepton number transport equations. After taking angular
moments of conservative formulations of the general relativistic Boltzmann
equation, we specialize to a conformally flat spacetime, which also serves as
the basis for four further limits. Two of these---the multidimensional special
relativistic case, and a conformally flat formulation of the spherically
symmetric general relativistic case---are given in appendices for the sake of
comparison with extant literature. The third limit is a weak-field,
`pseudo-Newtonian' approach \citep{kim_etal_2009,kim_etal_2012} in which the
source of the gravitational potential includes the trace of the stress-energy
tensor (rather than just the mass density), and all orders in fluid velocity
$v$ are retained. Our primary interest here is in the fourth limit:
`$\mathcal{O}(v)$' moment equations for use in conjunction with Newtonian
self-gravitating hydrodynamics. We show that the concept of `$\mathcal{O}(v)$'
transport requires care when dealing with both conservative four-momentum and
conservative lepton number transport, and present two self-consistent options:
`$\mathcal{O}(v)$-plus' transport, in which an $\mathcal{O}(v^2)$ energy
equation combines with an $\mathcal{O}(v)$ momentum equation to give an
$\mathcal{O}(v^2)$ number equation; and `$\mathcal{O}(v)$-minus' transport, in
which an $\mathcal{O}(v)$ energy equation combines with an $\mathcal{O}(1)$
momentum equation to give an $\mathcal{O}(v)$ number equation.
