  Recently, the formalism needed to relate the finite-volume spectrum of
systems of nondegenerate spinless particles has been derived. In this work we
discuss a range of issues that arise when implementing this formalism in
practice, provide further theoretical results that can be used to check the
implementation, and make available codes for implementing the three-particle
quantization condition. Specifically, we discuss the need to modify the upper
limit of the cutoff function due to the fact that the left-hand cut in the
scattering amplitudes for two nondegenerate particles moves closer to
threshold; we describe the decomposition of the three-particle amplitude
$\mathcal{K}_\text{df,3}$ into the matrix basis used in the quantization
condition, including both $s$ and $p$ waves, with the latter arising in the
amplitude for two nondegenerate particles; we derive the threshold expansion
for the lightest three-particle state in the rest frame up to
$\mathcal{O}(1/L^5)$; and we calculate the leading-order predictions in chiral
perturbation theory for $\mathcal{K}_\text{df,3}$ in the $\pi^+\pi^+K^+$ and
$\pi^+K^+K^+$ systems. We focus mainly on systems with two identical particles
plus a third that is different ("2+1" systems). We describe the formalism in
full detail, and present numerical explorations in toy models, in particular
checking that the results agree with the threshold expansion, and making a
prediction for the spectrum of $\pi^+\pi^+K^+$ levels using the two- and
three-particle interactions predicted by chiral perturbation theory.
