  We use our cluster population model, cBHBd, to explore the mass distribution
of merging black hole binaries formed dynamically in globular clusters. We
include in our models the effect of mass growth through hierarchical mergers
and compare the resulting distributions to those inferred from the third
gravitational wave transient catalogue. We find that none of our models can
reproduce the peak at $m_1\simeq 10M_\odot$ in the primary black hole mass
distribution that is inferred from the data. This disfavours a scenario where
most of the sources are formed in globular clusters. On the other hand, a
globular cluster origin can account for the inferred secondary peak at
$m_1\simeq 35M_\odot$, which requires that the most massive clusters form with
half-mass densities $\rho_{\rm h,0} \gtrsim 10^4 M_\odot \rm pc^{-3}$. Finally,
we find that the lack of a high mass cut--off in the inferred mass distribution
can be also explained by the repopulation of an initial mass gap through
hierarchical mergers. Matching the inferred merger rate above $\simeq
50M_\odot$ requires both initial cluster densities $\rho_{\rm h,0} \gtrsim 10^4
M_\odot \rm pc^{-3}$, and that black holes form with nearly zero spin. A
hierarchical merger scenario makes specific predictions for the appearance and
position of multiple peaks in the black hole mass distribution, which can be
tested against future data.
