Material dungeons are short climbs with enemies based on your highest cleared floor. Gather materials across three stages and face a boss with guards in the final stage.
Open "Material Dungeon" in "Dungeon" to see your tickets and the stage rewards. Prepare a party and depart from that screen; tickets refill as real time passes.
Enter to gather materials for revival or rebirth. Returning between stages preserves your haul, while a wipe loses the materials from that run. Check remaining health, since mercenaries face the same injury and death risks as in the tower.