Decarbonization of the electric grid has become an important world-wide priority and is actively happening in many ways by introducing innovations and new technologies from the generation sectors to the demand sectors. In particular, one promising pathway toward such net-zero carbon emissions is to utilize the demand-side flexibility with the increasing number of flexible loads in distribution networks. In this paper, we explore a load shifting strategy with the emerging concept of location marginal emissions (LMEs) to reduce carbon emissions. LMEs measure the impact of carbon emissions including the locational aspect in more granular way and thus provide a novel mechanism for the system operator and load aggregators to design the LME-based load shifting strategy, which can efficiently guide consumers and thus adjust their consumption behaviors. Simulation case studies on the IEEE test networks are performed to validate the capability of the proposed load shifting method to reduce carbon emissions. We also compare this with other relevant strategies to discuss multiple scenarios and corresponding results. While each provides a different level of flexibility, all the explored strategies tested have led to solutions that have lower carbon emissions, indicating the great potential of demand-side flexibility in reducing carbon emissions for future distribution networks.