An efficient Legendre-Galerkin method was applied to obtain a solution of the Maxwell dusty fluid flow past a changeable thickness stretching surface in porous medium. It is common to reduce a governing system of partial differential equations to a collection of ordinary differential equations using similarity transformation to obtain the asymptotic solutions using the Legendre-Galerkin technique. Comparison of the numerical results is made with previously published results under the special cases. The results are found to be in a good agreement. Numerical results were obtained for the skin-friction, velocity profiles of the fluid as well as velocity profiles of dusty particles for selected values of the governing parameters, such as velocity power index 𝑛, wall thickness α, elasticity β, fluid particle interaction βv, porosity γ, and mass concentration ℓ parameters. Also, the impacts of these parameter on the physical (surface heat flux and surface shear stress) and mechanical properties (surface cracking, strength, hardness, stiffness, etc.) are presented and investigated.