  Background: The LHCb Collaboration has studied a number of kinematic
correlations between $B$-hadron pairs through their subsequent decays to
$J/\psi$ pairs at 7 and 8 TeV for four minimum values of the $J/\psi$ $p_T$.
Purpose: In this work, these measurements are compared to calculations of $b
\bar b$ pairs and their hadronization and inclusive decays to $J/\psi J/\psi$
are compared to the same observables. Potential cold matter effects on the $b
\bar b$ pair observables are discussed to determine which are most likely to
provide insights about the system and why. Methods: The calculations, employing
the exclusive HVQMNR code, assume the same intrinsic $k_T$-broadening and
fragmentation as in [R. Vogt, Phys. Rev. C {\bf 98} (2018) 034907]. The pair
distributions presented by LHCb are calculated in this approach, both for the
parent $b \bar b$ and the $J/\psi J/\psi$ pairs produced in their decay. The
sensitivity of the results to the intrinsic $k_T$ broadening is shown. The
theoretical uncertainties due to the $b$ quark mass and scale variations on
both the initial $b \bar b$ pairs and the resulting $J/\psi$ pairs are also
shown. Possible effects due to the presence of the nucleus are studied by
increasing the size of the $k_T$ broadening and modification of the
fragmentation parameter. Results: Good agreement with the LHCb data is found
for all observables. The parent $b \bar b$ distributions are more sensitive to
the $k_T$ broadening than are the final-state $J/\psi$ pairs. Conclusions:
Next-to-leading order calculations with $k_T$ broadening, as in [R. Vogt, Phys.
Rev. C {\bf 98} (2018) 034907], can describe all correlated observables.
Multiple measurements of correlated observables are sensitive to different
nuclear effects which can help distinguish between them.
