This is only a limited answer - but for the smaller framelits/thinlets, this works -- I hadn't used my red sizzix w/ thin dies for a while - but I just now tried this again with several small thinlets & it worked great.
Cutting pad (the heavy white one that goes w/ the press machine)
paper/cardstock
Die - (facedown, obviously)
"Adapter"
You need to put that magnetic "System Converter" on the machine, then a quick press with the lever & I always get a perfect cut! This particular 'sandwich' is limited to the smaller dies that can fit within the space of the 'adapter'
I know at one time, I'd worked out a 'sandwich' with several cutting pads & (who knows what???) so that I could cut thin dies larger than that little adapter square. If I can figure that out again, I'll let you know! Basically, I think if you worked out some sort of thick shim, the same thickness as the adapter, it should work.
FYI - this is the "system converter" referred to above - it was made so you could use Sizzlets & embossing folders in the red machine:
Sizzix Machine Accessory - System Converter
And these are the 'adapters' they sold to use with thin dies - shame they never made larger ones:
Sizzix Accessory - Adapter, 10 Pack