This paper considers the equilibrium-free stability and performance analysis of discrete-time nonlinear systems. Two types of equilibrium-free notions are considered. Namely, the universal shifted concept, which considers stability and performance w.r.t. all equilibrium points of the system, and the incremental concept, which considers stability and performance between trajectories of the system. This paper shows how universal shifted stability and performance of discrete-time systems can be analysed by making use of the time-difference dynamics. Moreover, the existing results are extended for incremental dissipativity for discrete-time systems based on dissipativity analysis of the differential dynamics to more general state-dependent storage functions for less conservative results. Finally, it is shown how both these equilibrium-free notions can be cast as a convex analysis problem by making use of the linear parameter-varying framework, which is also demonstrated by means of an example.