This paper deals with homogenization of second order divergence form parabolic operators with locally stationary coefficients. Roughly speaking, locally stationary coefficients have two evolution scales: both an almost constant microscopic one and a smoothly varying macroscopic one. The homogenization procedure aims to give a macroscopic approximation that takes into account the microscopic heterogeneities. This paper follows "Diffusion in a locally stationary random environment" (published in Probability Theory and Related Fields) and improves this latter work by considering possibly degenerate diffusion matrices. The geometry of the homogenized equation shows that the particle is trapped in subspace of R^d.