Black hole binaries formed dynamically in globular clusters are believed to be one of the main sources of gravitational waves in the Universe. Here, we use our new population synthesis code, cBHBd, to determine the redshift evolution of the merger rate density and masses of black hole binaries formed in globular clusters. We simulate $\sim 2$ million models to explore the parameter space that is relevant to real clusters and over all mass scales. We show that when uncertainties on the initial cluster mass function and density are properly taken into account, they become the two dominant factors in setting the theoretical error bars on merger rates. Other model parameters (e.g., natal kicks, black hole masses, metallicity) have virtually no effect on the local merger rate density, although they affect the masses of the merging black holes. Modelling the merger rate density as a function of redshift as $R(z)=R_0(1+z)^κ$ at $z<2$, and marginalizing over uncertainties, we find: $R_0=7.2^{+21.5}_{-5.5}{\rm Gpc^{-3}yr^{-1}}$ and $κ=1.6^{+0.4}_{-0.6}$ ($90\%$ credibility). The rate parameters for binaries that merge inside the clusters are ${R}_{\rm 0,in}=1.6^{+1.9}_{-1.0}{\rm Gpc^{-3}yr^{-1}}$ and $κ_{\rm in}=2.3^{+1.3}_{-1.0}$; $\sim 20\%$ of these form as the result of a gravitational-wave capture, implying that eccentric mergers from globular clusters contribute $\lesssim 0.4 \rm Gpc^{-3}yr^{-1}$ to the local rate. A comparison to the merger rate reported by LIGO-Virgo shows that a scenario in which most of the detected black hole mergers are formed in globular clusters is consistent with current constraints, and requires initial cluster half-mass densities $\gtrsim 10^4 M_\odot \rm pc^{-3}$. Such models also reproduce the inferred primary black hole mass distribution for masses $13-30 M_\odot$, but under-predict the data outside this range.

Revised to include a comparison to the data from GWTC-2. Accepted in PRD; 23 pages, 13 figures