Models that are robust to aberrant choice behaviour have received limited attention in discrete choice analysis. In this paper, we analyse two robust alternatives to the multinomial probit (MNP) model. Both alternative models belong to the family of robit models, whose kernel error distributions are heavy-tailed t-distributions. The first model is the multinomial robit (MNR) model in which a generic degrees of freedom parameter controls the heavy-tailedness of the kernel error distribution. The second alternative, the generalised multinomial robit (Gen-MNR) model, has not been studied in the literature before and is more flexible than MNR, as it allows for alternative-specific marginal heavy-tailedness of the kernel error distribution. For both models, we devise scalable and gradient-free Bayes estimators. We compare MNP, MNR and Gen-MNR in a simulation study and a case study on transport mode choice behaviour. We find that both MNR and Gen-MNR deliver significantly better in-sample fit and out-of-sample predictive accuracy than MNP. Gen-MNR outperforms MNR due to its more flexible kernel error distribution. Also, Gen-MNR gives more reasonable elasticity estimates than MNP and MNR, in particular regarding the demand for under-represented alternatives in a class-imbalanced dataset.