In this paper, we present switched adaptive strategies to incrementally stabilise a class of Carathéodory nonlinear systems. Using the concept of switched matrix measures based on the use of multiple norms, and the transaction coefficients between these norms, sufficient conditions are presented to prove incremental stability of a switched system. Furthermore, based on these sufficient conditions, two strategies are proposed to incrementally stabilise a switched system of interest: one based on monitoring the switching signal in a continuous way, the other on monitoring it only at selected discrete-time instants. We prove that in both situations, the switched system can be effectively stabilised. As a representative application, we illustrate the efficiency of these strategies by synchronizing a network system with blinking (switching) graph structures.