Mobile-relay-assisted forwarding in delay-tolerant networks (DTNs) can improve network capacity and packet delivery ratio but, at the same time, may significantly increase energy consumption at the system level. In this paper, we propose a distance-based energy-efficient opportunistic forwarding (DEEOF) framework for the broadcast transmission in mobile DTNs. DEEOF strikes a balance between energy consumption and network performance by maximizing the energy efficiency while maintaining a high packet delivery ratio. In the proposed algorithms, we define the metric of the forwarding equivalent energy efficiency distance (FEED) for broadcast transmission to quantify the transmission distances achieving the same energy efficiency at different time instances, or with different numbers of the relays in the source's transmission radius. Based on the concept of FEED, we propose two DEEOF algorithms for opportunistic broadcast forwarding, which makes the forwarding decision by comparing the current energy efficiency with the estimated future expectation and distribution, respectively. The performance improvement of the proposed DEEOF algorithms is also demonstrated by simulation, particularly for the cases in which the source has very limited battery reserves.