Topological phases of matter connect mathematical principles to real materials, and may shape future electronic and quantum technologies. So far, this discipline has mostly focused on single-gap topology described by topological invariants such as Chern numbers. Here, based on a tunable kagome model, we observe non-Abelian band topology and its transitions in acoustic semimetals and insulators, in which the multi-gap Hilbert space plays a key role. In non-Abelian semimetals, the topological charges of band nodes are converted through the braiding of nodes in adjacent gaps, and their behaviour cannot be captured by conventional topological band theory. By controlling the geometry of the acoustic metamaterials, topological transitions are induced by the creation, annihilation, merging and splitting of band nodes. This reveals the underlying rules for the conversion and transfer of non-Abelian topological charges in multiple bandgaps. The resulting laws that govern the evolution of band nodes in non-Abelian multi-gap systems should inspire studies on multi-band topological semimetals and multi-gap topological out-of-equilibrium systems.

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