It is widely believed that bufferless crossbar switches with virtual-output queues (VOQ) at their inputs can only operate when their input-output connections are reconfigured in synchrony, i.e. only under fixed-size cell traffic. Packet-mode scheduling has been studied, but, again, assuming that all packets consist of an integer number of cells, where the scheduling time coincides with the cell time. We show that bufferless crossbars can operate directly on variable-size packets, with input-output connections being made and torn down asynchronously with respect to each other. Although such operation can initially be thought of as an extension of packet-mode scheduling, the critical difference is that now the scheduling time is much longer than packet-size granularity. We study a transformation of the well-known iSLIP scheduling algorithm to asynchronous mode of operation, and we show by simulation that it can be adapted to yield throughput close to 100% under a range of workloads. The overall result is an efficient scheduling operation, with the added advantages of eliminating (a) packet fragmentation overhead (no partially filled cells), and (b) packet reassembly in the egress datapath.