how is lily a dynamic character and what are the ways in which she evolves over the course of the book?
Keep in mind that a dynamic character is simply one who grows and changes throughout the book, rather than remaining stagnant. Lily, as a fourteen year old at the opening of the story, has much growing up to do, both as a result of her age and her circumstances.
The best way to fully answer this question, is to look and perhaps jot down some characteristics of Lily at the beginning, middle, and end of the novel. She starts out as a somewhat timid (but certainly not cowardly) girl who's sense of self is wrapped up in a dead mother and an emotionally distant father. In short, she does not feel loved by those around her, and therefore, does not love herself. She does, however, have the capacity to understand she can change her circumstances, and so she runs away with Rosaleen.
Lily changes while living at August's house, and though gradual, the changes are distinct. They could almost be tracked chapter-by-chapter. Chapter 12, however, is one of the biggest turning points (both in the story and in Lily's character). In a conversation with August, Lily finally admits her guilt about her mother's death, and inability to love herself. When August points out how many people do love her, it is as if a light finally penetrates the darkness that has been clouding her vision. She begins to build a confidence that can only come from self-love.
Finally, her changes are manifested in the final chapter of the book, when T. Ray arrives to bring Lily home. More than likely, had he appeared earlier, Lily would have timidly (and pitifully) gone with him, believing there was nothing she could do about it. However, given the changes in her heart and life, Lily is able to stand up to T. Ray and tell him she's going to stay with August. In this moment, Lily displays a sense of calm confidence and independence she possibly did not even know she had inside her, which could only be the result of her relationships and experience with August and the other other women who helped Lily define herself.