A family of velocity-based linearizations is proposed for a nonlinear system. In contrast to the conventional series expansion linearization, a member of the family of velocity-based linearizations is valid in the vicinity of any operating point, not just an equilibrium operating point. The velocity-based linearizations facilitate dynamic analysis far from the equilibrium operating points and enable the transient behaviour of the nonlinear system to be investigated. Using velocity-based linearizations, 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 linearization analysis is global in nature. The analysis techniques developed, although quite general, are motivated by the gain-scheduling design approach and have the potential for direct application to the analysis of gain-scheduled systems.