Inhomogeneous reionization acts as a source of arcminute-scale anisotropies in the cosmic microwave background (CMB), the most important of which is the kinetic Sunyaev–Zel’dovich (kSZ) effect. Observational efforts with the Atacama Cosmology Telescope (ACT) and the South Pole Telescope (SPT) are poised to detect this signal for the first time, with projected 1 μ K2-level sensitivity to the dimensionless kSZ power spectrum around a multipole of l= 3000, [Δl3000]2. Indeed, recent SPT measurements place a bound of [Δl3000 ]2 < 2.8 μ K2 at 95 per cent confidence level, which degrades to [Δl3000 ]2 < 6 μ K2 if a significant correlation between the thermal Sunyaev–Zel’dovich (tSZ) effect and the cosmic infrared background (CIB) is allowed. To interpret these and upcoming observations, we compute the kSZ signal from a suite of ≈100 reionization models using the publicly available code 21cmfast. Our physically motivated reionization models are parametrized by the ionizing efficiency of high-redshift galaxies, the minimum virial temperature of haloes capable of hosting stars, and the ionizing photon mean free path – a parametrization motivated by previous theoretical studies of reionization. We predict the contribution of patchy reionization to the l= 3000 kSZ power to be 1.5–3.5 μ K2. Therefore, even when adopting the lowest estimate in the literature for the post-reionization signal of , none of our models are consistent with the aggressive 2σ SPT bound that does not include correlations. This implies the following: (i) the early stages of reionization occurred in a much more homogeneous manner than suggested by the stellar-driven scenarios we explore, such as would be the case if, e.g. very high energy X-rays or exotic particles contributed significantly and/or (ii) that there is a significant correlation between the CIB and the tSZ. The later is perhaps not surprising, as massive haloes should host both hot gas and star-forming galaxies. On the other hand, the conservative SPT bound of [Δl3000 ]2≲ 6 μ K2 is compatible with all of our models and is on the threshold of constraining physically motivated reionization models. The largest patchy kSZ signals correspond to an extended reionization process, in which the sources of ionizing photons are abundant and there are many recombinations (absorptions in sinks). We point out that insights into the astrophysics of the early Universe are encoded in both the amplitude and shape of the kSZ power spectrum.