We evaluate the redshift distribution of binary black hole (BBH), black hole-neutron star binary (BHNS), and binary neutron star (BNS) mergers, exploring the main sources of uncertainty: star formation rate (SFR) density, metallicity evolution, common envelope, mass transfer via Roche lobe overflow, natal kicks, core-collapse supernova model, and initial mass function. Among binary evolution processes, uncertainties on common envelope ejection have a major impact: the local merger rate density of BNSs varies from ∼103 to ∼20 Gpc-3 yr-1 if we change the common envelope efficiency parameter from αCE = 7 to 0.5, while the local merger rates of BBHs and BHNSs vary by a factor of ∼2-3. The BBH merger rate changes by one order of magnitude, when 1σ uncertainties on metallicity evolution are taken into account. In contrast, the BNS merger rate is almost insensitive to metallicity. Hence, BNSs are the ideal test bed to put constraints on uncertain binary evolution processes, such as common envelope and natal kicks. Only models assuming values of αCE ≳ 2 and moderately low natal kicks (depending on the ejected mass and the supernovae mechanism), result in a local BNS merger rate density within the 90 per cent credible interval inferred from the second gravitational-wave transient catalogue.