For those who don't want to read a long guide:

Quick route: Basically wait until the enemy group to the south has passed the starting room. Then in Turn 3, get all the units out of the starting room and follow the enemy group with Asaga (and Lynn for extra reward) until you reach the room on the bottom right with the door to the right. This door will only be open in turn 8! Thats where you need to go to finish the mission.

If you want all rewards: You need to take all units with you while following the enemy group and take Cosette to open the room at the bottom left, to activate a switch. This switch will keep the goal room in the bottom right accessable, while you collect all the rewards with Lynn.

The following map shows all the rooms and their content at the start of the mission. The whole level consists of 15 rooms. Your goal is to reach room 15 with Asaga who is carrying Kayto and place him on the platform.

• Enemy group 1 (E1): Appears in turn 2 and moves to the left every turn. By turn 4 it will reach room 3 and stay there. This forces you to leave the room by turn 3.
  
• Enemy group 2 (E2): Circles through the rooms 1-9 counter-clockwise. It starts in room 3. It does not care about your position unless you are in the same room (which means game over).
  
• Enemy group 3 (E3): Appears as soon as the switch is activated. It will move down one room every turn until it reaches the room with the switch.
  
• The green dots are green crystals which can be collected by Lynn for 500 credits each (you can collect up to 4500 credits).
  
• The red passage ways can be open by Asaga, the grey ones can only be opened by the enemies.
  
• The violet passage can be opened permenantly by Cosette.
  
• The switch will turn all grey passages into red passages making them accessable with Asaga.

Now, before you make any move, you should enable "higher movement speed" in options -> difficulty -> costumize difficulty. This makes the mission less painful since otherwise you have to be very careful with your EN management and will be required to use Claudes gravity gun a lot. Enabling assists does not deny you any achievements and here it will decrease the pain so use it.

There is a quick way to end this mission for those who don't care about any credits and a slower way for those who want all the credits.

Quick route:

• Turn 1: Stay in room 9! Move Kryska and Cosette closer to the door to the south but don't go through the door!
  
• Turn 2: Stay in room 9! Enemy group E1 will appear and will reach room 9 in turn 4 (turn 5 as of V1.1.0.92).
  
• Turn 3: Move everyone into room 2 (to the south).
  
• Turn 4-7: Now let everyone stay in room 2 except for Asaga and Lynn. For the next turns, follow enemy group E2 with Asaga and Lynn (but don't be in the same room with them). Collect the rewards in room 4 with Lynn and let her stay there.
  It is important to stay close to the enemy group since the door to room 15 will close, when the enemy group E2 reaches room 3 again!
  
• Turn 8: Reach room 8 and stop following the enemy group E2. Move Asaga to room 15 and put Kayto down on the platform.

Slower route (collecting all credits)

• Turn 1: Stay in room 9! Move Kryska and Cosette closer to the door to the south but don't go through the door!
  
• Turn 2: Stay in room 9! Enemy group E1 will appear and will reach room 9 in turn 4 (turn 5 as of V1.1.0.92)..
  
• Turn 3: Move everyone into room 2 (to the south) and move them close to the door on the left (especially Cosette and Kryska).
  
• Turn 4-6: Follow the enemies until you reach room 6 (with the teleporter entry). Try to place Cosette and Claude close to the wall on the left at the end of Turn 6.
  
• Turn 7: Saw through the wall with Cosette, collect the credits with Lynn and press the switch with Claudes "Gravity Gun". This switch will turn all doors into doors which can be opened by Asaga (so the gray doors on the map turn to red ones). The switch also makes enemy group E3 appear. They will reach the room with the switch in 2 turns. Asaga should further follow the enemy into room 7.
  
• Turn 8: By Turn 8 Asaga reaches room 8. Teleport Lynn to room 8. Make sure to get Claude and Cosette out of room 12, otherwise they will be killed by enemy group E3.
  
• Turn 9: Move Asaga and Lynn up to room 5 and collect the credits.
  
• Turn 10: Move Asaga and Lynn up to room 3 and collect the credits.
  
• Turn 11-12: Now move Asaga back to room 8 and open room 15. Go in there and place Kayto on the podest to finish the mission. If you cannot make it by turn 12, make sure to move the other units out of room 6!

I hope this guide is helpful. If you have questions or if you see any mistakes in my guide please feel free to contact me here or on the discord server (nickname: Uncle Neppi)