  We present a model-independent, non-perturbative relation between
finite-volume matrix elements and infinite-volume
$\textbf{0}\rightarrow\textbf{2}$ and $\textbf{1}\rightarrow\textbf{2}$
transition amplitudes. Our result accommodates theories in which the final
two-particle state is coupled to any number of other two-body channels, with
all angular momentum states included. The derivation uses generic, fully
relativistic field theory, and is exact up to exponentially suppressed
corrections in the lightest particle mass times the box size. This work
distinguishes itself from previous studies by accommodating particles with any
intrinsic spin. To illustrate the utility of our general result, we discuss how
it can be implemented for studies of
$N+\mathcal{J}~\rightarrow~(N\pi,N\eta,N\eta',\Sigma K,\Lambda K)$ transitions,
where $\mathcal{J}$ is a generic external current. The reduction of rotational
symmetry, due to the cubic finite volume, manifests in this example through the
mixing of S- and P-waves when the system has nonzero total momentum.
