Skyrim:Cleansing the Stones

Quick Walkthrough
# Use the Bend Will shout to cleanse the Beast Stone.
# Use the Bend Will shout to cleanse the Earth Stone.
# Use the Bend Will shout to cleanse the Sun Stone.
# Use the Bend Will shout to cleanse the Water Stone.

Detailed Walkthrough
With the Wind Stone cleansed, return to Skaal Village and tell Storn Crag-Strider the good news. He will suggest you to do the same for the remaining four stones of Solstheim. He doubts it will fully stop whatever Miraak is doing, but it may slow his progress. Your map will be updated and show you the location of your targets: Beast Stone, Earth Stone, Sun Stone, and Water Stone.

Beast Stone
The Beast Stone is southwest of Skaal Village; this is the nearest stone you may find. Two entranced Rieklings are working there. A leveled cultist is patrolling around the stone and monitoring the work of Miraak's poor slaves. The cultist will attack you on sight, so defeat your enemy and use the Bend Will shout to cleanse the stone. The structure around the stone will collapse and a leveled lurker will raise from the water. The two Rieklings won't turn hostile to you. They will stay here for a short moment then return to their tribe at Thirsk Mead Hall.

Sun Stone
Keep traveling south and you'll find the Sun Stone in the north of Tel Mithryn. A group of Reavers are focusing on their building. Shout at the stone and a leveled lurker will attack you. The fight will be much easier with the help of these Reavers. They won't attack you after the combat; they'll just wander around the stone and give you some generic, non-hostile responses. After you leave this area, they will be removed from the game.

Earth Stone
Travel to Raven Rock and you should find the Earth Stone without much trouble. Bralsa Drel, Rirns Llervu, and three Redoran guards have lost themselves here, doing their work mindlessly. Cleanse the stone and two lurkers will appear this time. One is near the stone, the other one is near the coastline. The guards will try their best to give you a hand. Pay close attention to the safety of Bralsa as she gives a quest later on. They will return to the city soon after regaining their freedom.

Water Stone
Walk along the west coast of Solstheim to find the Water Stone. The shrine at this stone is being worked on by a group of sailors, and a cultist is monitoring their work. The cultist will attack you without hesitation. Defeat your enemy and do the same to the stone. A leveled lurker will rise from the water and try to kill you. Be careful and watch the sky, as you may attract the attention of a low-level dragon nearby. If you do, prepare for a chaotic battle. Fortunately, you don't need to guarantee the safety of every sailor. They will return to their boat docked in the bay once they are freed. After you leave this area, they will sail back to Skyrim with their ship.

Notes
Before you cleanse the stones, the entranced people working there will repeat the following texts mechanically: "Here in his shrine.(Here in his temple.) That they have forgotten. Here do we toil. That we might remember. By the night we reclaim.(Here we reclaim.) What by day was stolen.(What faithless minds have stolen.) Far from ourselves. He grows ever near to us. Our eyes once were blinded. Now through him do we see. Our hands once were idle. Now through them does he speak. And when the world shall listen. And when the world shall see. And when the world remembers. That world will cease to be."
The entranced Rieklings around the Beast Stone will repeat some similar texts in their own language: "Wafoo athaaaal. Bertiflik a goroowala. Hra woong aba. Tika twafaara filik. Wilberk tikflaaar. Yuchaaa tungar cha tiweki. Bura warag faaanig. Tiri fuklik taaar aburak."
More people and Rieklings would appear at Beast Stone, Sun Stone, and Water Stone during the night.
The Tree Stone is the only All-Maker Stone that is not cleansed by the end of this quest. 
This quest must be completed to finish The Gardener of Men and the rest of the expansion's main questline.
Prior to completing this quest, there is a chance that you will wake up at one of the stones whenever you sleep. See the notes section on the All-Maker Stones article for more detail.
You can cleanse the stones before the quest has started. If you cleanse all of them before getting the Dragonborn quest, Storn will send you to find Frea at the Tree Stone, instantly completing Dragonborn and starting The Temple of Miraak.

Bugs
 
Travel to the Wind Stone and acquire its power to get the quest working properly. Stones you've already tried to cleanse will spawn Lurkers as soon as you near them but the shrines will not explode.
Alternatively, simply leaving the cell and returning to it, or saving and reloading the saved game, may resolve the problem, with the Lurker(s) appearing and the quest updating after the fight.
 

Quest Stages