I made a punch very similar to that which Pat DeWitt describes.
I have most tubular punches, from very small to about 1-1/2" in 1/64"
increments in the small sizes, and in 1/16" increments in the larger
sizes, but I never got the size needed for Ampico pouches so I turned
one from some decent steel that is hard enough for light use.

Instead of hammering, I use a press as it is gentler on the punch.
Pace Punches has round dies in a great variety of sizes and at good
prices.  I made the one for Ampico only because I have just about every
other die that I will ever need and didn't want to make up a minimum
order or pay excess shipping.  Anyone who does much cutting or round
shapes should check out Pace and get a full set.

For punching washers, I made a little punchy thing with a slide and
two holders with bearings that allow them to be slid back and forth
to punch the ID first and then the OD.  After making that I got a CNC
milling machine and now I let the machine do the punching while I do
other things.

I was able to make 3/64" ID 3/16" OD washers quite accurately that the
professionals wanted an obscene price to make unless I bought 100,000
pieces.  To make washers I wrote a simple program to produce the 'g'
codes to make a dense grid for any punch diameter and any sheet size.
The simple program ask for a few parameters and makes the file.  Much
easier than setting it up in a CAM program.  If anyone does it the same
way and wants the program, just ask.

Another very affordable possibility for the larger diameter punches is
to make steel rule dies.  These are made with a pre-sharpened thin
piece of hardened and tempered steel called steel rule.  You can buy
these from Pace or probably McMaster Carr, etc.

They are not good for small diameter dies or complex shapes unless you
know what you are doing, but for a simple shape anyone can make them --
you just cut inside and outside pieces of quality plywood (or MDFB if
you are cheap) and set the rule between the in and outside pieces.

A thin band or jig saw blade or even a coping saw (expertly used) can
make both the in and out pieces at once.  For round dies you can use a
hole saw or an adjustable hole cutter.  Then epoxy the rule between the
pieces and glue onto a backing of either hard board or metal.

The easiest way to use the dies if you don't have a press is to put
the stock on the die and press or gently hammer a block on the top.
End-grain fine grain wood works well, as Pat says, but the "self healing"
plastic cutting sheet found in craft and fabric stores also makes a good
cutting surface.  You can also use an arbor press or a drill press just
as a pressing device (but not with it running!).

Best regards,
Spencer Chase