We present systematic theoretical investigations to explore the microscopic mechanisms leading to the formation of antiferromagnetism in $\mathrm{R}{\mathrm{u}}_2\mathrm{Mn}Z(Z=\mathrm{Sn},\mathrm{Sb},\mathrm{Ge},\mathrm{Si})$ full Heusler alloys. Our study is based on first-principles calculations of interatomic Mn-Mn exchange interactions to set up a suitable Heisenberg spin model and on subsequent Monte Carlo simulations of the magnetic properties at finite temperature. The exchange interactions are derived from the paramagnetic state, while a realistic account of long-range chemical disorder is made in the framework of the coherent potential approximation. We find that in the case of the highly ordered alloys ($Z=\mathrm{Sn}$ and Sb), the exchange interactions derived from the perfectly ordered $L{2}_1$ structure lead to Néel temperatures in excellent agreement with the experiments, whereas, in particular in the case of Si, the consideration of chemical disorder is essential to reproduce the experimental Néel temperatures. Our numerical results suggest that by improving a heat treatment of the samples to suppress the intermixing between the Mn and Si atoms, the Néel temperature of the Si-based alloys can potentially be increased by more than 30%. Based on calculated biquadratic exchange couplings, we evidence a lifting of degeneracy of the antiferromagnetic ground states on a frustrated face-centered-cubic lattice in the fully ordered compounds. Furthermore, we show that in strongly disordered $\mathrm{R}{\mathrm{u}}_2\mathrm{MnSi}$ alloys, a distinct change in the antiferromagnetic ordering occurs.

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