I use brake cleaner / carb cleaner spray and compressed air. Some shops have a heated air assisted flushing system to clean out material. Many make do with a electric fuel pump , filter and a tank of solvent.

Cleaning a cooler is difficult because there are many places for metal / clutch material to get stuck. It is a good idea to install an inline filter on the outlet side of the cooler. If you can't run the filter directly inline, use a loop of hose and splice it in there.