  We study the inclusive production of hadrons with bottom flavor at the LHC
and its luminosity upgrade. We describe the collinear fragmentation of singly
$b$-flavored hadrons, $B$ mesons and $\Lambda_b$ baryons, via the KKSS07
determination of fragmentation functions, while for charmed $B$ mesons,
$B_c(^1S_0)$ and $B_c(^3S_1)$ particles, we employ the novel ZCFW22 set, built
on the basis of state-of-the-art nonrelativistic QCD inputs. We use the JETHAD
multimodular working environment to analyze rapidity and transverse-momentum
distributions for observables sensitive to the associated emission of two
hadrons or a hadron-plus-jet system. Our reference formalism is the NLL/NLO$^+$
hybrid collinear and high-energy factorization, where the standard collinear
description is improved by the inclusions of energy logarithms resummed up to
the next-to-leading approximation and beyond. We provide a corroborating
evidence that $b$-flavor emissions act as fair stabilizers of the high-energy
resummation, thus serving as valuable tools for precision studies of
high-energy QCD. As a bonus, we highlight that the predicted production-rate
hierarchy between noncharmed $b$-hadrons and charmed $B_c(^1S_0)$ mesons is in
line with recent LHCb estimates. This serves as simultaneous benchmark both for
the hybrid factorization and the single-parton fragmentation mechanism from the
nonrelativistic QCD effective theory.

High Energy Physics - Phenomenology;High Energy Physics - Experiment