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, $delta_{pipi}$, 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.
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