Once you get the hang of aligning prints to both sides of a chipboard mat (tips in a future blog post), you just align the punch and it's hammer-time. I've read that some people recommend doing it in as few hammer strikes as possible, but I couldn't really ever get through in fewer than eight, and everything looks fine on my side. One thing I would personally recommend would be to put the self-healing mat on a HARD surface (concrete), not a table, otherwise some of the force from the hammer blow will be transmitted to the table and reduce the effectiveness of the punch.

Have you used arch punches in other creative ways? I can imagine making non-circular chits by overlapping punches, and oval-punches using specific oval drive-punches (they're up to about 1/2", so maybe not for actual game-pieces)...

UPDATE: User Murm3l on reddit asked how they work on foamcore. Since I had some on hand I tried it out: