So the ZX Spectrum Next team have added a Z80 DMA chip, the Z8410 chip to be more precise. This was popular add on in Europe ( more info here: http://velesoft.speccy.cz/data-gear.htm ), and has some very cool advantages for future games, especially because it's now a standard part. However information on it is a little thin on the ground, so after some major googling, I've found the info on them.
The DMA port can be 11 or 107, and it can transfer around 865k per second
From the web page above: Max. speed of data transfer on ZX128+ is...
17.3 kB(17727 bytes) / frame = 865.6 kB(886350 bytes) / second.
Or.... 1 byte every 4 T-States.
Now here is the really cool bit, because it's tied to the clock speed, when you speed up the NEXT (as it can go 7Mhz, 14Mhz and 28Mhz), the DMA will also speed up! This is amazing, as it means you'll be able to transfer 1.73Mb/s, 3.46Mb/s, and an astounding 6.92Mb/s! At 28Mhz you won't be using Layer 2, and that means you could copy the original Layer 1 screen in just 13.5 scanlines, which is incredible.
The registers are below, and here is the Datasheet