The most common solutions are to have a human replace the document or an easel full of clipped documents. You can fit quite a few greeting cards side by side on some of our larger plot-area machines.
There are automated solutions. Some are scrappy (like an inkjet placed in front of each AxiDraw to print a blank page in front of it), some are expensive (like a robot arm to move the paper), some are full custom, and some are using off-the-shelf paper handling equipment.