The foundation of the stack is the assembly of pneumatic decks and striking fingers.
As previously mentioned, this assembly can be separated as a whole from the secondary valve chest, and then broken down into its constituent components.

While this series of operations is not rocket surgery it is surely among the longest and dirtiest jobs of the entire restoration: you have a lot of work ahead of you!

Before starting the teardown in earnest, it’s a good idea to take photographs from different angles and different distances, to document how the parts looked before you took them all apart.
And a reminder to make note of key measurements such as span and overhang of the striker pneumatics, the average position of the leather nuts on the striker finger lift rods, and indexing all support blocks (position/orientation) before removing them from the decks.

The first order of business is to disengage and remove the striker finger lift rods from the pneumatic hangers, or whatever similar arrangement is necessary in your particular action.

This usually involves removing the lower leather nut from the threaded lift rod, to then free it from the metal hanger or wood finger. If the nuts are extremely old, they may be crushed with pliers. If they are moderately old, you should be able to back them off with a simple custom tool that can be made with thin metal tube stock. John Tuttle has instructions here, and this is how my version came out:

Since you are in the teardown stage, there is not an enormous amount of care needed to remove the nuts. By contrast, when it comes time to replace new nuts on your finished pneumatic decks, extreme care must be taken!

But let us not get ahead of ourselves, for now just get them off as best as can. If corrosion has set in at the joint of the nut and rod, you will likely need pliers to break the joint and get them started.

Once that is done, and the rods are disconnected, if it is possible to do so, disconnect the rods from the striker fingers, and store them, keeping them in order if possible. A sheet of boxboard will suffice. They can be cleaned and polished later.

Turn your attention to the striker fingers. Number them, make note of any damage or stripped screw holes in the deck, as well as the condition of the bushings, and then remove them.
If the fingers have a glue joint to the deck underneath, a firm but careful blow with a block of the same thickness should free them with minimal damage.

Set aside the fingers and screws, the screws do not really need to be kept in order, as they are all theoretically the same. Like the rods, the fingers will need a good cleaning and lubrication before remounting.

Finally, if there is a finger rest rail along the top of the pneumatics, make note of measurements and orientation; index it if necessary, then remove it from the deck. Be aware that there may be screws hidden under the layer of felt atop the rail. Remove the felt but make note of the thickness of this material, as it will save much trouble to duplicate it as closely as possible.