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Multistage buffered switching fabrics are the most efficient method for scaling packet switches to very large numbers of ports. The Benes network is the lowest-cost switching fabric known to yield operation free of internal blocking. Backpressure inside a switching fabric can limit the use of expensive off-chip buffer memory to just virtual-output queues in front of the input stage. This article extends the known credit-based flow control (backpressure) architectures to the Benes network. To achieve this, we had to successfully combine per-flow backpressure, multipath routing (inverse multiplexing), and cell resequencing. We present a flow merging scheme that is needed to bring the cost of backpressure down to O(N) per switching element, and for which we have proved freedom from deadlock for a wide class of multipath cell distribution algorithms. Using a cell-time-accurate simulator, we verify operation free of internal blocking, evaluate various cell distribution and resequencing methods, compare performance to that of ideal output queuing, the iSLIP crossbar scheduling algorithm, and adaptive and randomized routing, and show that the delay of well-behaved flows remains unaffected by the presence of congested traffic to oversubscribed output ports.