Can the studio live 32i or any studio live be used to mix down recordings from a DAW?

Can the studio live 32i or any studio live be used to (physically) mix down recordings from a DAW? For that matter is there any digital desk that allows the desk to physically mix the recording from the computer on the desk itself? I think the Mackie 1640i will do this (I know the 1640i is an analog desk)
I'm not talking about multiple AD/DA converters and lots of patching to accomplish this. And I'm not talking about mega $50,000 desks. Maybe? $10,000 or less? Thanks.

Are you asking about a "controller" or just asking if you can use the mixer as a playback surface... because you can run however many channels your desk has ... back through the mixer and mix that way...

So, the 32Ai you would be able to mix down at least 32 tracks, the 16.4.2, 16 tracks and so on...Controllers would allow you to mix infinite tracks down, by selecting which tracks you have selected in the controllers..

This would be like having the tracks on a multitrack reel to reel, and mixing down through a desk... only the reel is a DAW and the desk is a mixer like the SL...or Ai...

Thanks. So the Ai has a

A built-in 48x34 FireWire S800 interface (is FWS800 different from FW800?) So I can record a live show to S1 and take the rig home and mix it still retaining...say...20 individual tracks down thru the Ei using the knobs, encoders, and sliders and put the refined 20 tracks back into S1 for mix down to 2 tracks or/and use the desk(Ai) to refine the tracks and record stereo (2 tracks) back to the computer? I guess I'm asking can the StudioLive mixers be used as DAW control surfaces? Sorry if these are stupid questions, thanks again.

If the sl32 works like the other sl's then you can't go daw-->desk-->daw on all channels.

You can only go daw-->desk--> 2track (that can be returned to the daw).