This study theoretically and experimentally explores the behaviour of axisymmetric gravity currents of Newtonian and power-law fluids in inhomogeneous porous media. Systematic heterogeneity along the vertical is represented by a power-law permeability variation governed by the parameter ω, mimicking trends in natural media. A self-similar solution describing (i) the rate of propagation and (ii) the profile of the current is derived by considering a current of volume proportional to time raised to a non-negative power α. Four critical values of α are determined: the first two affect the time dependency of the radius, height and average gradient of the current on flow behaviour index n and ω; the third one dictates if the current accelerates or decelerates; the fourth one governs the asymptotic validity of the thin current approximation. Experimental validation is performed using shear-thinning suspensions and Newtonian mixtures in constant- and variable-flux regimes. A stratified porous medium composed of four uniform strata of glass beads with different diameters is used for this purpose. The experimental results for the radius and profile of the current agree well with the self-similar solution except at the beginning of the process, due to the limitations of the 1-D assumption and to boundary effects near the injection zone. An uncertainty analysis indicates that the rheological fluid behaviour and the variation in permeability significantly affect the propagation of the current.