Years ago I bought some non-stick material for lining my baking tins. In fact it's pretty much the same stuff that's used now for the non-stick brown craft mats. Anyway, I had some odd scraps so I tried die-cutting with it and left it in the die. The next time I cut some card it almost jumped out of the die on its own! After putting it through a few times the liner was nicely set into the die and I find that even the subsequent embossing is fine. I use fairly lightweight card for die-cutting so not sure how well it would work with thicker card.
Having used up all my tin-lining scraps I wasn't about to cut up my non-stick craft mat, so I headed off to one of those famous Pound shops. They sell the non-stick sheets. They are much thinner and therefore perhaps even better for my purpose than my original thicker liner.

I have to say, to cover myself, that it is NOT recommended by the die or die-cutting machine manufacturers and, like me, you would use it at your own risk.