In the dynamics of Biot poroelastic materials, the fluid flow is not affected by the deformation of the solid elastic frame. In contrast, in permeable materials whose solid stiff frames have flexible thin flat films attached, i.e. permeo-elastic materials, the fluid flow can be significantly modified by the presence of the films. As a consequence of the local fluid–film interaction, and in particular of the local resonances, the classical local physics is changed and departs from that leading to the Biot description. In this paper, the two-scale asymptotic homogenisation method is used to derive the macroscopic description of sound propagation in air-saturated permeo-elastic materials. This description is asymptotically analysed to determine the conditions for which the geometrical and mechanical properties of the films strongly affect the effective properties of the material. The developed theory is illustrated numerically and validated experimentally for a prototype material, evidencing the atypical acoustic behaviour.