Categorize settlers as either combatants (usually guards, provisioners, and scavengers), non-combatants (e.g.farmers and merchants), or "ignored" (unassigned settlers) at armory containers, and gear will be distributed from their associated containers.
You can sort armor into these containers by hand, or use the included armor sorting functionality (it uses AWKCR keywords for the sorting, so if you want to distribute armors that aren't supported by AWKCR you'll need to sort them by hand.)Version 2.1 adds flags to settlers that indicate that armor or weapons have been distributed to them.
Central Settlement Armory steps through all of the settlers in your settlement, or all connected settlements, retrieving weapons and ammo from their inventory and moving them to junk storage.
It then selects a weapon from the armory container, more or less at random, and equips it on the settler, also supplying one of the appropriate ammunition (this will give them infinite ammo for that weapon).
I've tested this with about 34 connected settlements with 300 settlers, and it takes a couple of minutes to distribute the gear.
It will take longer the more settlements and settlers you have, but progress can be checked by activating the Armory Telestorage Manager during distribution. It works the same as weapon distribution, but has a container for each armor slot (clothing, one piece armor sets, underarmor, helmets, chest pieces, left arm, right arm, left leg, right leg).