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Free-surface thin-film flows can principally be described by two types of models. Lubrication models assume that shear stresses are dominant in the force balance of the flow and are appropriate where there is little or no slip at the base of the flow. Conversely, membrane or ‘free-film’ models are appropriate in situations where there is rapid slip and normal (or extensional) stresses play a significant role in force balance. In some physical applications, notably in glaciology, both rapid and slow slip can occur within the same fluid film. In order to capture the dynamics of rapid and slow slip in a single model that describes the entire fluid film, a hybrid of membrane and lubrication models is therefore required. Several of these hybrid models have been constructed on an ad hoc basis in glaciology, where they are usually termed ‘higher order models’. Here, we present a self-consistent asymptotic analysis of the most …