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This paper deals with a discrete diffusive predator–prey system involving two competing preys and one predator in a shifting habitat induced by the climate change. By applying Schauder's fixed‐point theorem on various invariant cones via constructing several pairs of generalized super‐ and subsolutions, we establish four different types of supercritical and critical forced extinction waves, which describe the conversion from the state of a saturated aboriginal prey with a pair of invading alien predator–prey, two competing aboriginal coexistent preys with an invading alien predator, a pair of aboriginal coexistent predator–prey and an invading alien prey, and the coexistence of three species to the extinction state, respectively. Meanwhile, the nonexistence of some subcritical forced waves is showed by contradiction. Furthermore, some numerical simulations are given to present and promote the theoretical results. [ABSTRACT FROM AUTHOR]