Which screens
Backpack, workbench, chemistry station, cement mixer, and campfire. The forge is unchanged. It has its own material slots.
Craft from the claim
Players only
While a crafting screen is open, the recipe list, the have counts, and the craft button also count items in player-placed chests inside the land claim you are standing in.
If your backpack is short, the missing materials move from those chests into your backpack first, then the normal craft or upgrade runs.

Backpack, workbench, chemistry station, cement mixer, and campfire. The forge is unchanged. It has its own material slots.
Stackable materials. Tools, weapons, and armor are never taken. A recipe that also needs a quality item still takes the stackable parts from the chests. A gyro's wheels can come from a chest. The small engine stays in your backpack.
On a multiplayer server the mod waits for the server to confirm each chest change. If that chest is in use, nothing is crafted or upgraded, and anything already taken stays in your backpack.
Upgrading a block with a repair tool uses the same chests for wood, cobblestone, concrete, steel, and the rest.
With the Upgrade Bench mod installed, that bench's material cost comes from the same chests. The two mods you put in the bench still have to be in the bench.
UpgradeFromChests, and it is off by default. Set it to on in settings.txt to get the 1.1.x behavior back. ModInfo, site descriptions, and the README how-it-works paragraph match: crafting uses claim chests; upgrades only when that setting is on.