The mechanical strength of a polycrystalline material can be drastically weakened by a phenomenon known as grain boundary (GB) premelting that takes place, owing to the so-called disjoining potential, when the dry GB free energy sigma(gb) exceeds twice the free energy of the solid-liquid interface sigma(sl). While previous studies of GB premelting are all limited to equilibrium conditions, we use a multi-phase field model to analyze premelting dynamics by simulating the steady-state growth of a liquid layer along a dry GB in an insulated channel and the evolution of a pre-melted polycrystalline microstructure. In both cases, our results reveal the crucial influence of the disjoining potential. A dry GB transforms into a pre-melted state for a grain-size-dependent temperature interval around T-m, such that a critical overheating of the dry GBs over T-m should be exceeded for the classical melting process to take place, the liquid layer to achieve a macroscopic width, and the disjoining potential to vanish. Our simulations suggest a steady-state velocity for this transformation proportional to sigma(gb) - 2 sigma(sl). Concerning the polycrystalline evolution, we find unusual grain morphologies and dynamics, deriving from the existence of a pre-melted polycrystalline equilibrium that we evidence. We are then able to identify the regime in which, due to the separation of the involved length scales, the dynamics corresponds to the same curvature-driven dynamics as for dry GBs, but with enhanced mobility.