MrMark:
I don't think there's a way to send a value to a port and twiddle individual bits on control lines as might be needed to service an external parallel RAM all under the DMA controller.
With a DUE, you can read continuously or not, via an AHB DMA, the content of PIO_PDSR (up to 32 bits = 32 pins), or continuously or not write, via an AHB DMA, to PIO_ODSR (up to 32 bits = 32 pins).