In this work, we study the problem of routing packets between undifferentiated sources and sinks in a network modeled by a multigraph. We provide a distributed and local algorithm that transmits packets hop by hop in the network and study its behaviour. At each step, a node transmits its queued packets to its neighbours in order to optimize a local gradient. This protocol is thus greedy since it does not require to record the history about the past actions, and lazy since it only needs informations of the neighborhood. We prove that this protocol is stable in the sense that the number of packets stored in the network stays bounded as soon as the sources injects a flow that another method could have exhausted. In particular, our protocol stays stable even if the feasibility condition is not strict on topologies with several sources and destinations, under a conjecture when the value of the flow is constrained at the destination nodes. We therefore reinforce a result from the literature that worked for differentiated suboptimal flows.