There are now several approaches to compactifying the moduli space of plane curves: GIT, K-moduli, KSBA, etc. These methods can be unified via "wall-crossing", by viewing a plane curve as a divisor with some varying coefficient in P2.  We will review this picture through the example of quartics, and discuss how we can build upon these ideas to compute the cohomology and Chow rings of these moduli spaces. This talk is based on joint work in progress with Donggun Lee.