It is not possible to obtain the gear by killing Mourners after Plague's End, as they all disappear and are replaced by Elves. Instead, a replacement set can be found by inspecting the chest in the Head Mourner's office in West Ardougne.
Note: if you plan on teleporting after this battle your Magic level will be reduced so without a restore potion, the only teleports you will be able to cast are home teleport or Mobilising Armies teleport. Go to Arandar (the overpass that you use to enter the elven lands), with 7 inventory spaces free, and kill a Mourner on the western side of the mountain (in Arandar, north of Lletya). When you start the battle, the Mourner will throw a potion at you that will reduce all of your combat stats including life points and prayer. You should still be able to kill the Mourner if you have a good weapon. Also note that if you one-hit the mourner, your stats will not be reduced, so long as the mourner doesn't attack you.
During the battle with the Mourner in Mourning's End Part I, the Mourner trousers will be torn and the Mourner top stained with blood. These items obviously cannot be worn, and will require some repairing and cleaning. Later (after starting Within the Light), additional Mourner gear can be obtained without the need to repair the trousers or top.
The gear must be worn in order to access the Mourners' basement headquarters in West Ardougne to get to the Dark beasts, runite ore, and the east entrance of the Temple of Light to the Death altar. Wearing the gear also unlocks unique dialogue with the citizens in West Ardougne and certain NPCs in East Ardougne. However, the Mourner gear CANNOT be used to enter the incinerator area north of West Ardougne.
After completion of Plague's End, the mourner gear may be reclaimed from the mourner headquarters in West Ardougne. To obtain, enter the headquarters and descend the trapdoor. In the south eastern room is a chest from which the gear may be reclaimed.
Most Mourners actually wear a plague jacket, which has gloves attached to it (they are much longer and more visible), and the plague trousers and the leggings of this set have no visual difference when worn.
While some people in the twin cities of East Ardougne and West Ardougne change their dialogue if the player wears the complete outfit, as does Lord Iorwerth in the Elf Camp; others do not. Although this gear is meant to conceal the player's identity, to the point where others cannot tell if the player is elf or human, it appears that nevertheless some seem to be aware that it is the player who is wearing mourner gear, while others are not.
When Arianwyn wears the Mourner gear (during Within the Light) the cape seems to have a different colour scheme, with the inside of the cape being red. This is only when he's still standing in Lletya though, and not when he accompanies the player through the temple.
In the house across from the bank on Lunar Isle, there is a mourner shirt hanging on the north wall.