The relaxation kinetics of three-dimensional lattice model proteins with Gō potential is studied. A kinetic transition from an exponential behavior to a nonexponential one with a denatured native spinodal is characterized. The transition temperatures T(k), obtained from simulations and a semiquantitative estimation, are found to be the same. The change in free energy landscape during the transition is discussed microscopically by studying the detailed folding processes of various paths. The connection of T(k) with the foldability is also studied by a Z-score-like quantity T(f)/T(k).