As noted above, there are two new material types needed for Awakening Evolution. These are "Awakening Crystals" and "Book of Awakening" and they come in Red, Blue, and Green varieties.
The most common way to obtain these materials is to run the special Challenge Dungeons, where they have a chance to drop when you clear the Dungeon (higher difficulties offer a better chance for a drop). The Awakening Materials may also sometimes be obtained through Special Campaigns or Gachas.
With recent updates to the Challenge Dungeons, you can now earn Points when you beat them. These can then be used to buy specific Awakening Materials.
The "Book of Awakening" can be obtained from any Challenge Dungeon, while the Awakening Crystals are specific to certain Challenge Dungeons.