An analytical model is proposed for the propagation of elastic waves in inhomogeneous substrates containing inhomogeneous inclusions (considering inhomogeneities in the modulus and density) based on elastic wave propagation in a homogeneous medium. The proposed method provides novel concepts and serves as a theoretical reference for determining the dynamic stress concentration and interface displacement in functionally graded material (FGM) composites. An inhomogeneous substrate containing inhomogeneous circular inclusions under the action of SH waves is taken as an example: the variations in the modulus and density are considered to have an exponential form for the inclusions and a power law form for the substrate, and the dynamic stress response in the substrate induced by the inclusions is analyzed. Calculation examples are used to determine the effects of the reference wavenumber in the substrate, the inclusion-to-substrate wavenumber ratio and the inhomogeneity in the substrate and inclusions on the dynamic stress concentration factor (DSCF) distribution at the periphery of the inclusions. The dynamic stress concentration induced by the inclusions is found to be sensitive to changes in the substrate heterogeneity. Increasing the inhomogeneity parameter of the substrate results in a sharp increase in the dynamic stress concentration around a circular inclusion. Finally, regulating the heterogeneity of the inclusions can reduce the dynamic stress concentration to some extent.