  In this work we develop a Lorentz-covariant version of the previously derived
formalism for relating finite-volume matrix elements to $\textbf 2 + \mathcal J
\to \textbf 2$ transition amplitudes. We also give various details relevant for
the implementation of this formalism in a realistic numerical lattice QCD
calculation. Particular focus is given to the role of single-particle form
factors in disentangling finite-volume effects from the triangle diagram that
arise when $\mathcal J$ couples to one of the two hadrons. This also leads to a
new finite-volume function, denoted $G$, the numerical evaluation of which is
described in detail. As an example we discuss the determination of the $\pi \pi
+ \mathcal J \to \pi \pi$ amplitude in the $\rho$ channel, for which the
single-pion form factor, $F_\pi(Q^2)$, as well as the scattering phase,
$\delta_{\pi\pi}$, are required to remove all power-law finite-volume effects.
The formalism presented here holds for local currents with arbitrary Lorentz
structure, and we give specific examples of insertions with up to two Lorentz
indices.
