This paper investigates distributed link selection (LS) for a multi-hop buffer-aided relay network consisting of one source, one destination, and multiple relays, where each node has access only to local instantaneous channel state information (CSI) and the buffer status of its adjacent nodes. In particular, by improving the alternate-transmission strategy, we propose a novel flexible distributed LS scheme that flexibly selects odd- and even-numbered links in each time slot. Notably, acquisition of local buffer-state information is integrated into the distributed LS agreement process, so it incurs no extra signaling overhead. We also derive the average throughput and packet delay of the proposed scheme by constructing a two-layer Markov model and enumerating all feasible buffer-state transitions. In addition, a simplified expression and an asymptotic analysis are provided to give further insight into the average throughput. Theoretical analysis and simulations show that the proposed flexible scheme substantially outperforms a baseline alternate scheme and closely approaches the performance of a related centralized LS scheme.