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This paper extends the high-order entropy stable (ES) adaptive moving mesh finite difference schemes developed in Duan and Tang (2022) to the two- and three-dimensional (multi-component) compressible Euler equations with the stiffened equation of state (EOS). The two-point entropy conservative (EC) flux is first constructed in the curvilinear coordinates, which is nontrivial in the case of the stiffened EOS. With the aid of the high-order discretization of the geometric conservation laws and the linear combinations of the two-point EC fluxes, the high-order semi-discrete EC schemes are derived. The high-order semi-discrete ES schemes are constructed by adding suitable high-order dissipation term to the EC schemes such that the semi-discrete entropy inequality is satisfied and unphysical oscillations are suppressed. The high-order dissipation term is built on the multi-resolution weighted essentially non-oscillatory (WENO) reconstruction and the newly derived scaled eigenvector matrices. The explicit strong-stability-preserving Runge–Kutta methods are used for the time discretization and the mesh points are adaptively redistributed by iteratively solving the mesh redistribution equations with an appropriate monitor function, which is adapted to the multi-component flow and encodes more physical characteristics of the solutions. Several 2D and 3D numerical tests are conducted on the parallel computer system with the MPI programming to validate the accuracy and the ability to effectively capture the localized structures of the proposed schemes. [ABSTRACT FROM AUTHOR]