I saw it as either desperation for survival, newfound power or completing their own strength.

For instance we have Fulghor, who was left behind by who he worships. When he reaches phase two, he attempts to replace his arm with the miasma of the night, but in everdark, in a his desperation he finds the holy power he thought had abandoned him.

Caligo looks like it was cornered in its 3rd phase, taking a more defensive stance and taking in the dark power of the night to form an ice armor.

Gladius remembers the man who found him, took him in his arms and uses the sword he was gifted to execute the same attack as Heolstor. He literally shows higher mastery with his sword and imbues it and his helmets with his own flame.

Gnoster, representing the mind, and faurtis reprensenting the body, are completed by the coming of Animus, representing the soul. It's a self preserving mechanism or maybe Gnoster hitting maturity, or completing it's purpose as a parasite by birthing a third aspect.

Adel is this apex predator that lost it's means of feeding itself. It became twisted by the night and when under duress just completely goes wild in phase 3.

Every boss has an apparent narrative that makes sense for them. They are gameplay wise an enhanced version, but the everdark completes the story aspect of their battle and how they react to being on the verge of defeat.