Soret or thermal diffusion is known to impact combustion mixtures by diffusion of species along a temperature gradient. In this numerical modelling study we show that Soret diffusion can cause considerable local changes in the stabilization of H2 enriched premixed flames. First we validate two reduced Soret diffusion models with a more complete multi-component model. One model is able to capture the effect of Soret diffusion in the flame zone while the other model captures the effect in the flame zone as well as in the regions where high temperature gradients might exist e.g., near hot burner, side walls. Using the two reduced models along with conjugate heat transfer, we analyze bluff-body stabilized flames and the effect of Soret diffusion in the flame zone and near hot walls. For such flames, heat transfer between the flame and the burner results in local quenching. This heat is recirculated through the burner towards the unburnt gases creating a temperature gradient in the boundary layer around the burner. Because of Soret diffusion, this gradient drives H2 and even CH4 towards the burner walls in the unburnt mixture thus creating a local stratification (in the mixture approaching the flame base). As a result of this stratification, the burning rate of the flame near the anchoring location is enhanced, which causes more heat to be lost to the burner. The flame stabilizes at a new location where the strain rate is higher in order to neutralize the enhancement in burning rate and a new balance between flame and flow velocity is found. The effect of this local stratification is shown to be profound only when there is considerable heat loss to the burner such as for flames near flashback.