The plasma behaviour in an air-propelled Hall thruster (SITAEL HT5k) is investigated by means of numerical modelling and experimental measurements. The air plasma employed (nitrogen/oxygen mixture) is firstly chemically characterized, identifying the most relevant species produced: four neutrals (N₂, N, O₂, O), four positive ions (N₂⁺, N⁺, O₂⁺, O⁺) and electrons. Then, it is explored the kinetic theory of ion-ion streaming instabilities in the presheath, relevant to electropositive multispecies plasmas, extending it to the four-ions case under study. These results are used to adapt a non-stationary 1D fully fluid Hall thruster model to operation with an air plasma. Calibration with an experimental discharge current signal confirms the physical validity of the formulation in reproducing breathing mode oscillations, even in multispecies plasmas. The experimental data also consist of fast-diving triple Langmuir probe measurements, which are used to validate the model, showing reasonable accuracy of the spatio-temporal distributions of the plasma properties.