In the framework inspired by R. L. Hughes model (Transp. Res. B, 2002) for pedestrian evacuation in a corridor, we establish existence of a solution by a topological fixed point argument. This argument applies to a class of models where the dynamics of the pedestrian density ρ (governed by a discontinuous-flux Lighthill,Whitham and Richards model ρt + (sign(x − ξ(t))ρv(ρ)) x = 0) is coupled via an abstract operator to the computation of a Lipschitz continuous "turning curve" ξ. We illustrate this construction by several examples, including the standard Hughes' model with affine cost, and either with open-end conditions or with 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 general costs.