This study addresses the stability issue of a class of fractional-order switched non-linear systems. A general stability condition is established based on which two new concepts named Mittag–Leffler increment and average Mittag–Leffler increment are proposed and two alternative stability conditions are obtained. These conditions fully reveal the trade-off between each subsystem's dynamics, and the value of the fractional order. The new results are further applied to a class of fractional-order multi-agent systems with switching cooperative control laws.