Microscopic calculations of four-body collisions become very challenging in the energy regime above the threshold for four free particles. The neutron-${}^3$He scattering is an example of such process with elastic, rearrangement, and breakup channels. We aim to calculate observables for elastic and inelastic neutron-${}^3$He reactions up to 30 MeV neutron energy using realistic nuclear force models. We solve the Alt, Grassberger, and Sandhas (AGS) equations for the four-nucleon transition operators in the momentum-space framework. The complex-energy method with special integration weights is applied to deal with the complicated singularities in the kernel of AGS equations. We obtain fully converged results for the differential cross section and neutron analyzing power in the neutron-${}^3$He elastic scattering as well as the total cross sections for inelastic reactions. Several realistic potentials are used, including the one with an explicit $Delta$ isobar excitation. There is reasonable agreement between the theoretical predictions and experimental data for the neutron-${}^3$He scattering in the considered energy regime. The most remarkable disagreements are seen around the minimum of the differential cross section and the extrema of the neutron analyzing power. The breakup cross section increases with energy exceeding rearrangement channels above 23 MeV.
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