Markov chains are natural models for a wide range of applications, such as financial and environmental time series. For risk assessment, it is the extreme events that are of most practical concern and hence, it is critical to understand the behaviour of the chain within an extreme event. For asymptotically dependent Markov chains existing formulations fail to capture the full evolution of the extreme event when the chain moves out of the extreme tail region and for asymptotically independent chains recent results fail to cover well-known asymptotically independent processes such as Markov processes with a Gaussian copula between consecutive values. We use more sophisticated limiting mechanisms that cover a broader class of asymptotically independent processes than current methods and reveal features which existing methods reduce to a degenerate form associated with non-extreme states. Our results extend to higher order Markov processes and are used to motivate a flexible conditional extreme value model.