  We describe a general framework for analyzing orbits of systems containing
compact objects (neutron stars or black holes) in a class of Lagrangian-based
alternative theories of gravity that also admit a global preferred reference
frame. The framework is based on a modified Einstein-Infeld-Hoffmann (EIH)
formalism developed by Eardley and by Will, generalized to include the
possibility of Lorentz-violating, preferred-frame effects. It uses a
post-Newtonian $N$-body Lagrangian with arbitrary parameters that depend on the
theory of gravity and on "sensitivities" that encode the effects of the bodies'
internal structure on their motion. We determine the modified EIH parameters
for the Einstein-{\AE}ther and Khronometric vector-tensor theories of gravity.
We find the effects of motion relative to a preferred universal frame on the
orbital parameters of binary systems containing neutron stars, such as a class
of ultra-circular pulsar-white dwarf binaries, the amplitudes of the effects
depend upon "strong-field" preferred-frame parameters $\hat{\,\,\alpha}_1$ and
${\hat{\,\,\alpha}}_2$, which we relate to the fundamental modified EIH
parameters. We also determine the amplitude of the "Nordtvedt effect" in a
triple system containing the pulsar J0337+1715 in terms of the modified EIH
parameters.
