Flame propagation in compositionally stratified methane-air mixtures was studied experimentally as a function of the equivalence ratio distribution in the unburnt mixture. Stratification was established in a controlled manner using a convective-diffusive balance in a very slow fuel-air mixture flow in an optically accessible test chamber. The flame speed was shown to be significantly higher than the one corresponding to a homogeneous mixture of the local equivalence ratio for mixture compositions close to the lean flammability limit. Also a significant extension of the lean flammability limit was observed. It was established that the local spatial gradient of the equivalence ratio was not sufficient to describe the departure of stratified combustion from quasi-homogeneity. Instead, an appropriately defined integral parameter that depended on the history of flame propagation was shown to determine when the flame could not be treated as a series of premixed flamelets propagating at the local adiabatic flame speed.