This review deals with the structural transitions arising in anion-deficient Mn- and Cu-based perovskites when changing the anion content. Special attention is devoted to the local structure of such materials, studied with electron diffraction and high-resolution electron microscopy. Using examples of three-dimensional or layered manganites and cuprates, it is shown that oxidation/reduction reactions in such compounds are accompanied by the formation of domain structures. Fragmentation into domains can have intrinsic reasons, related to the loss of symmetry elements due to the ordering of anion vacancies, or because of the simultaneous realization of several different anion ordering patterns with close formation energies. When extra anions are inserted at low temperatures, limited diffusion ability also results in the appearance of domains of different structures arising from a local inhomogeneity in the anion distribution. The relevance of the investigation of the local structure for a correct structure interpretation is discussed.