The response of laminar methane/air counterflow premixed flames under sinusoidal equivalence ratio oscillation was investigated numerically. The timescales of the oscillation were chosen to be sufficiently longer than the flame timescale so that the flame responds quasi-steadily. The response of periodically stratified flame (SF) with a detailed reaction mechanism exhibited the "back-support" effect, in that the consumption speed S-c response deviated increasingly from S-c of steady homogeneous flames (HFs) at higher oscillation frequencies. It was shown that even when the imposed oscillation timescale is much longer than the flame timescale, the flame response can still be delayed under a sufficiently large equivalence ratio gradient. Subsequently, the above results were compared with those obtained with a global four-step mechanism that omits back-diffusion radicals into the reaction zone. As a result, SFs with the global mechanism displayed a much smaller back-support effect in both lean and rich mixtures. Further analysis with modified diffusion coefficients revealed the dominant roles of H-2 and radical species diffusion in inducing the back-support effect. Contrary to the previous findings, variations in burned gas temperature were found to play a negligible role in modifying S-c. Additionally, the hysteresis of the back-support effect under periodical stratification was found to be more prominent on the richer side because of the presence of a larger H-2 pool. (C) 2019 The Combustion Institute. Published by Elsevier Inc. All rights reserved.