Diffusion auction is an emerging business model where a seller aims to incentivise buyers in a social network to diffuse the auction information thereby attracting potential buyers. We focus on designing mechanisms for multi-unit diffusion auctions. Despite numerous attempts at this problem, existing mechanisms either fail to be incentive compatible (IC) or achieve only an unsatisfactory level of social welfare (SW). Here, we propose a novel graph exploration technique to realise multi-item diffusion auction. This technique ensures that potential competition among buyers stay “localised” so as to facilitate truthful bidding. Using this technique, we design multi-unit diffusion auction mechanisms MUDAN and MUDAN-m. Both mechanisms satisfy, among other properties, IC and 1/m-weak efficiency. We also show that they achieve optimal social welfare for the class of rewardless diffusion auctions. While MUDAN addresses the bottleneck case when each buyer demands only a single item, MUDAN-m handles the more general, multi-demand setting. We further demonstrate that these mechanisms achieve near-optimal social welfare through experiments.