Pedestrian evacuation in a corridor can de described mathematically by different variants of the model introduced by R. L. Hughes (Transp. Res. B, 2002). We identify a class of such models for which existence of a solution is obtained via a topological fixed point argument. In these models, the dynamics of the pedestrian density ρ (governed by a discontinuous-flux Lighthill, Whitham and Richards model ρ_t + (sign(x − ξ(t))ρv(ρ))_x = 0) is coupled to the computation of a Lipschitz continuous “turning curve” ξ. We illustrate this construction by several examples, including the Hughes model with affine cost (a variant of the original problem that is encompassed in the framework of El Khatib et al. (ZAMP, 2013)). Existence holds either with open-end boundary conditions or with boundary conditions corresponding to panic behaviour with capacity drop at exits. Other examples put forward versions of the Hughes model with inertial dynamics of the turning curve and with general costs.