High-temperature QCD can be described, through dimensional reduction, by a 3D effective field theory. Electrostatic QCD consists of a gauged-scalar theory, while magnetostatic QCD of a pure 3D gauge theory. The use of dimensionally reduced effective theories to predict 4D observables relies on the matching between the 3D and 4D theories, which is known only perturbatively. Given the known issues with perturbation theory at finite temperature it is important to perform a non-perturbative study. We have devised a strategy to perform a fully non-perturbative computation of the matching using gradient flow quantities in a finite volume setup. In this talk I will explain how we perform the matching over a large range of temperatures with high precision while keeping all lattice systematics under control. As a first step in this study, we investigate the matching between the finite temperature pure gauge theory and its 3D counterpart for which I report the preliminary results.