High order methods play important roles in the modelling of compressible multi-component flows. However, they may generate negative sound speed, which leads to an instability of the numerical schemes. In this paper, we propose bound- and positivity-preserving limiters for high order finite difference schemes, based on which the equilibriums of the velocity and pressure are preserved throughout the whole computation of contact moving interface problems with the ideal and stiffened equations of state. For illustration purpose, high order alternative WENO scheme is taken for example. Numerical examples verify the theory and demonstrate the robustness of the proposed bound- and positivity-preserving limiters.