Each DMA channel uses the AXI master interface to read data descriptor tables and read/write data buffers. The DMA implements round-robin arbitration. Arbitration is never granted to any request if the common buffer does not have enough space. Consequently, the DMA does not put back pressure on the AXI read channel.
If there is not enough space in the common buffer, the arbiter stays parked on the requesting channel until space is available.