Each mercenary carries one weapon, one armor and one trinket. What you pick up on a floor goes to the warehouse, and you fit it from the Gear tab of the mercenary sheet.
Tap "Swap" on a slot to see only the gear that fits it, best first, each row showing the power difference against what they wear now. Following that number alone never loses you anything, and "Auto-equip all" in the Stash does the comparison for the whole guild.
Tiers climb with the floors. The deeper the drop, the higher the tier, and 7 tiers are open so far. Every generation opens one more.