Third-generation (3G) gravitational-wave (GW) detectors will be able to observe binary-black-hole mergers (BBHs) up to redshift of $sim 30$. This gives unprecedented access to the formation and evolution of BBHs throughout cosmic history. In this paper we consider three sub-populations of BBHs originating from the different evolutionary channels: isolated formation in galactic fields, dynamical formation in globular clusters and mergers of black holes formed from Population III (Pop III) stars at very high redshift. Using input from populations synthesis analyses, we created two months of simulated data of a network of 3G detectors made of two Cosmic Explorers and an Einstein Telescope, consisting of $sim16000$ field and cluster BBHs as well as $sim400$ Pop III BBHs. First, we show how one can use non-parametric models to infer the existence and characteristic of a primary and secondary peak in the merger rate distribution. In particular, the location and the height of the secondary peak around $zapprox 12$, arising from the merger of Pop III remnants, can be constrained at $mathcal{O}(10%)$ level. Then we perform a modeled analysis, using phenomenological templates for the merger rates of the three sub-population, and extract the branching ratios and the characteristic parameters of the merger rate densities of the individual formation channels. With this modeled method, the uncertainty on the measurement of the fraction of Pop III BBHs can be improved to $lesssim 10%$, while the ratio between field and cluster BBHs can be measured with an uncertainty of $sim 50%$.
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