Summary: We investigate the off-equilibrium dynamics of a classical spin system with \(O(n)\) symmetry in \(2<D<4\) spatial dimensions and in the limit \(n\rightarrow\infty \). The system is set up in an ordered equilibrium state and is subsequently driven out of equilibrium by slowly varying the external magnetic field \(h\) across the transition line \(h_{\text c}=0\) at fixed temperature \(T\leqslant T_{\text c}\). We distinguish the cases \(T=T_{\text c}\) where the magnetic transition is continuous and \(T<T_{\text c}\) where the transition is discontinuous. In the former case, we apply a standard Kibble-Zurek approach to describe the non-equilibrium scaling and formally compute the correlation functions and scaling relations. For the discontinuous transition we develop a scaling theory which builds on the coherence length rather than the correlation length since the latter remains finite for all times. Finally, we derive the off-equilibrium scaling relations for the hysteresis loop area during a round-trip protocol that takes the system across its phase transition and back. Remarkably, our results are valid beyond the large-\(n\) limit.