There is a need to radically increase mobility of terrain vehicles through new modalities of vehicle locomotion, i.e., by establishing a new technological paradigm in vehicle dynamics and mobility. The new paradigm greatly applies to military vehicles for the radical improvement of tactical and operational mobility. This article presents a new technological paradigm of agile tire slippage dynamics that is studied as an extremely fast and exact response of the tire-soil couple to (i) the tire dynamic loading, (ii) transient changes of gripping and rolling resistance conditions on uniform stochastic terrains and (iii) rapid transient changes from one uniform terrain to a different uniform terrain. Tire longitudinal relaxation lengths are analyzed to characterize the longitudinal relaxation time constants. A set of agile characteristics is also considered to analyze agile tire slippage dynamics within a time interval that is close to the tire longitudinal relaxation time constants. The presented paradigm of agile tire slippage dynamics lays out a foundation to radically enhance vehicle terrain mobility by controlling the tire slippage in its transient phases to prevent the immobilization of a vehicle. Control development basis and requirements for implementing an agile tire slippage control are also analyzed and considered.