  Recently, the LIGO-Virgo collaborations have reported the coalescence of a
binary involving a black hole and a low-mass gap object (LMGO) with mass in the
range $\sim2.5-5M_\odot$. Such detections, challenge our understanding of the
black hole and neutron star mass spectrum, as well as how such binaries evolve
especially if isolated. In this work we study the dynamical formation of
compact object pairs, via multiple binary-single exchanges that occur at the
cores of globular clusters. We start with a population of binary star systems,
which interact with single compact objects as first generation black holes and
LMGOs. We evaluate the rate of exchange interactions leading to the formation
of compact object binaries. Our calculations include all possible types of
binary-single exchange interactions and also the interactions of individual
stars with compact object binaries that can evolve their orbital properties,
leading to their eventual merger. We perform our calculations for the full
range of the observed Milky Way globular cluster environments. We find that the
exchanges are efficient in forming hard compact object binaries at the cores of
dense astrophysical stellar environments. Furthermore, if the population size
of the LMGOs is related to that of neutron stars, the inferred merger rate
density of black hole-LMGO binaries inside globular clusters in the local
Universe is estimated to be about $0.1 \, \text{Gpc}^{-3}\text{yr}^{-1}$.
