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We study the time-dependent heat equation on its space-time domain that is discretised by a k-spacetree. k-spacetrees are a generalisation of the octree concept and are a discretisation paradigm yielding a multiscale representation of dynamically adaptive Cartesian grids with low memory footprint. The paper presents a full approximation storage geometric multigrid algorithm for this setting that combines the smoothing properties of multigrid for the equations's elliptic operator with a multiscale solution propagation in time. While the runtime and memory overhead for tackling the all-in-one space-time problem is bounded, the holistic approach promises to exhibit a better parallel scalability than classical time stepping, adaptive dynamic refinement in space and time fall naturally into place, on-time computational steering is eased as the algorithm delivers guesses for the solutions's long-term behaviour immediately, and, finally, backward problems benefit from the the solution being available for any point in space and time.