  Hierarchical black hole (BH) mergers are one of the most straightforward
mechanisms to produce BHs inside and above the pair-instability mass gap. Here,
we investigate the impact of globular cluster (GC) evolution on hierarchical
mergers, and we account for the uncertainties related to BH mass pairing
functions on the predicted primary BH mass, mass ratio and spin distribution.
We find that the evolution of the host GC quenches the hierarchical BH assembly
already at the third generation, mainly due to cluster expansion powered by a
central BH sub-system. Hierarchical mergers match the primary BH mass
distribution from GW events for $m_1 > 50 \, \mathrm{M_{\odot}}$, regardless of
the assumed BH pairing function. At lower masses, however, different pairing
functions lead to dramatically different predictions on the primary BH mass
merger rate density. We find that the primary BH mass distribution evolves with
redshift, with a larger contribution from mergers with $m_1 \geq 30 \,
\mathrm{M_{\odot}}$ for $z\geq{}2$. Finally, we calculate the mixing fraction
of BBHs from GCs and isolated binary systems. Our predictions are very
sensitive to the spins, which favor a large fraction ($>0.6$) of BBHs born in
GCs, in order to reproduce misaligned spin observations.
