The analysis and homogenization of a heat conduction problem with moving boundary for a two-phase medium is considered. The medium in question is assumed to be highly heterogeneous with a high contrast in the heat conductivities. In this context, the normal velocity governing the motion of the interface separating the two competing phases is assumed to be prescribed. Parametrizing the boundary motion via a height function, the so-called Direct Mapping Method is employed to construct a coordinate transform characterizing the changes with respect to the initial setup of the geometry. Using this transform, well-posedness of the problem is established. After characterizing the limit behavior (with respect to the heterogeneity parameter) of the functions related to the transformation, the corresponding homogenized problem is deduced.