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We introduce a numerical workflow to model and simulate transient close-contact melting processes based on the space-time finite element method. That is, we aim at computing the velocity at which a forced heat source melts through a phase-change material. Existing approaches found in the literature consider a thermo-mechanical equilibrium in the contact melt film, which results in a constant melting velocity of the heat source. This classical approach, however, cannot account for transient effects in which the melting velocity adjusts itself to equilibrium conditions. With our contribution, we derive a model for the transient melting process of a planar heat source. We iteratively cycle between solving for the heat equation in the solid material and updating the melting velocity. The latter is computed based on the heat flux in the vicinity of the heat source. The motion of the heated body is simulated via the moving mesh strategy referred to as the Virtual Region Shear-Slip Mesh Update Method, which avoids remeshing and is particularly efficient in representing unidirectional movement. We show numerical examples to validate our methodology and present two application scenarios, a 2D planar thermal melting probe and a 2D hot-wire cutting machine.