We study the nucleation and propagation of the normal zone in large composite superconductors, considering the relatively long time of current redistribution in the stabilizer. We propose a model treating the composite as an effective electrical circuit, which yields two diffusion equations for the electric current and temperature distributions along the conductor. Numerical simulations are performed to study the dynamics of propagating normal domains in cryostable conductors. We derive an analytical solution for the margins of existence and velocity of propagation of the domain. The effect of the boiling crisis on the dynamics of the normal zone is also studied.