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The present study is concerned with the dynamic response of an anisotropic composite structure due to a normal moving load on its irregular rough surface. The composite structure is comprised of an irregular incompressible heterogeneous transversely isotropic fluid-saturated poroelastic layer lying over a transversely isotropic substrate. The mathematical formulation of this structure gives rise to a boundary value problem with specified boundary conditions, and the perturbation method has been used to tackle the irregular surface problem. The expressions for the induced shear and normal stresses in layer and substrate of the composite structure are derived analytically in closed form due to the moving load. As a special case of the problem, the deduced expressions of the induced stresses are validated with the pre-established and standard results. The effect of several substantial parameters such as vertical depth, heterogeneity parameter, porosity parameter, frictional coefficient, irregularity depth, and irregularity factor on the induced shear as well as normal stresses of the layer and substrate has been delineated graphically by the numerical computation. Moreover, a comparative study of the various types of irregularity, namely rectangular irregularity, parabolic irregularity and no irregularity (regular boundary surface) on the induced shear and normal stresses in the layer and, substrate, is carried out by means of graphs, and some considerable peculiarities are outlined.