Not sure I follow. That may be true with most all-in-one boxes (AW4416, Akai DP24, etc), but standalone digital consoles are another matter.
Each time a sample rate is changed on the DM, it's rebooted for each chosen type. The firewire card drives the ship, so to speak, and the DAW slaves up to the DM's S/R (sample RATE) - the speed with which each wave portion is captured into bits 'n bytes.
Bit Depth - or 'WordLength' - references the 'stack' of 0s (zeros) and 1s (ones). The difference among these is about available dynamic range and 'foot-room'. That's why it's always best to track at 24bit (or 32 with the 8 additional 0's) to avail yourself of the signal to noise benefits over 16bit. The latter, of course, is the general release spec - ie: - CD/Redbook 16/44.1 - or the smaller cousins - MP3 - ie: 16/44.1 - and the various 'Fraunhoffer' compression choices relative to KBS (kilobytes per second). The more KBS, the higher the resolution (and detail) of the file.
CaptDan