A family of velocity-based linearisations is proposed for a nonlinear system. In contrast to the conventional series expansion linearisation, a member of the family of velocity-based linearisations is valid in the vicinity of any operating point, not just an equilibrium operating point. The velocity-based linearisations facilitate dynamic analysis far from the equilibrium operating points and enable the transient behaviour of the nonlinear system to be investigated. Using velocity-based linearisations, stability conditions are derived for both smooth and non-smooth nonlinear systems which avoid the restrictions, to trajectories lying within an unnecessarily, perhaps excessively, small neighbourhood about the equilibrium operating points, inherent in existing frozen-input theory. For systems where there is no restriction on the rate of variation, the velocity-based linearisation analysis is global in nature. The analysis techniques developed, whilst quite general, are motivated by the gain-scheduling design approach and have the potential for direct application to the analysis of gain-scheduled systems.