Surely the artist didn't sit there copying their cartoon onto every single copy, but I don't understand how these cartoons could have been reproduced on each copy otherwise.
What /u/CptBuck said. Also, cartoons don't really appear in newspapers until the 19th century, first in the form of lithographs; Punch in Britain and La Caricature and Le Charivari in France in the 1830s. Before that they exist as stand alone sheets. What we would recognize as political cartoons, employing caricature, don't really appear until around the 1770s in Britain, printed as etchings, usually hand-colored.
By no means an expert on this but as far as I'm aware the most common techniques included etching, wherein an acid was applied to a metal surface which was then used as a printing plate and woodcuts or engraving which were done by carving the image into wood or metal, respectively. A later technique was lithography.