We examine the nature of the transition to the antiferromagnetically ordered state in the half-filled three-dimensional Hubbard model using the dual-fermion multiscale approach. Consistent with analytics, in the weak-coupling regime we find that spin-flip excitations across the Fermi surface are important and that the strong-coupling regime is described by Heisenberg physics. In the intermediate-interaction, strong-correlation regime we find aspects of both local and nonlocal correlations. We analyze the critical exponents of the transition in the strong-coupling regime and find them to be consistent with Heisenberg physics down to an interaction of U/t=10.