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We consider the flow control problem for a general class of compartmental nonlinear systems, which can be associated with a graph whose nodes represent subsystems with their own internal dynamics, and whose arcs represent flow links among them. We consider a network-decentralized control: each agent controls a link between two nodes and decides its actions based on the states of these nodes only. We first provide general necessary and sufficient stabilizability conditions, proving that suitable network-decentralized strategies assure robust stability. We also show that, if all the subsystems at the nodes are marginally stable, a proper network-decentralized strategy asymptotically assures the minimum-norm flow, without requiring communication among agents.