This work investigates the sound absorption properties of permeable metamaterials which include acoustic resonators and multiscale sorptive porous media as key building-block elements. The macroscopic description of acoustic wave propagation in multiscale sorptive metamaterials is established using the two-scale method of homogenisation. This upscaling procedure elucidates the key role of pressure-activated mass sources on the atypical effective compressibility of the studied class of metamaterials. The results are exemplified by introducing analytical models which are successfully compared with numerical predictions. Furthermore, particular attention is paid on to how inner resonances and diffusion processes affected by nanoscale phenomena can be exploited to engineer efficient low-frequency sound absorbers.