Your magnet is losing its magnetic properties because of the pressure from the die cutting machine. The magnets we use in day to day life have atoms with electrons that act like tiny little bar magnets, or "spins", that are all lined up in the same direction and add up to one big bar magnet. However, they usually don't start out this way. In magnetic materials, the spins like to line up with each other in regions called "domains", but each domain might not be lined up with a neighboring domain. This results in a lump of material that wouldn't stick to a fridge because if you add up the magnetism from all the domains, they are pointing in different directions and add to nothing.

You can align these domains in the same direction by passing the material through a really strong magnetic field since all the spins want to point in the same direction as the field. But if you hit the magnet, or in your case squeeze it through a die cutter, the mechanical shock will cause the domains to lose their alignment or split into multiple non-aligned domains, which results in a loss of magnetic properties.

When a magnet exerts a force on an object, especially if that force generates an electric current, the object exerts an equal and opposite force (Newton's third law) back on the magnet. If the force imparted by magnet does work, the response force behaves in a fashion that drains the energy stored in the magnet that creates the magnetic field that the magnet exerts. Because I don't know how cutting dies work or even really what they are, I can't explain how your cutting dies are demagnetizing your magnets, but the general principle is most likely that they are extracting the energy from them as a result of the magnets doing work on them, somehow.

Yes, I've noticed that my sheet magnets weaken with time and use. Sheet magnets aren't particularly strong, and yours sound like they're getting heavy use. Rigid magnets get weaker when you drop them. Electrons are well aligned in magnets.*

*In ferromagnetic substances, electrons align readily (in regions called magnetic domains) with outside magnetic fields, such as the Earth's magnetic field, and stay that way. Paramagnetic substances also align with outside magnetic fields, but the effect is both weaker and more fleeting. From this site