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Multistage packet switches that feature a limited amount of buffers in the switching fabric and distribute most of their buffering capacity over the port cards have recently gained popularity due to their scalability properties and flexibility in supporting quality-of-service (QoS) guarantees. In such switches, the replication of multicast packets typically occurs in-the fabric. This approach minimizes the amount of resources needed to sustain the internal expansion in traffic volume due to multicasting, but also exposes multicast flows to head-of-line (HOL) blocking in the ingress port cards. The distributed scheduler that arbitrates the transfer of packets through the switch and the flow-control scheme that restricts access to the fabric buffers have the most critical role in safeguarding the QoS guarantees of multicast flows against HOL blocking. We add minimal overhead to a well-known QoS framework for multistage packet switches to define the generalized distributed multilayered scheduler (G-DMS), which achieves full support of QoS guarantees for both unicast and multicast flows. The main novelty of the G-DMS is in the mechanism that regulates access to the fabric buffers, which combines selective backpressure with the capability of dropping copies of multicast packets that violate the negotiated profiles of the corresponding flows.