In recent years K-stability has successfully constructed K-moduli spaces for Fano varieties and log Fano pairs. A natural next direction in this topic is the explicit description of these K-moduli spaces in specific examples. The study of K-moduli of log del Pezzo pairs formed by a del Pezzo surface of degree d and an anti-canonical divisor is of particular importance, as they are the first natural lower-dimensional examples of such explicit descriptions. These moduli spaces naturally depend on one parameter, providing a natural problem in variations of K-moduli spaces, by exhibiting wall-crossing phenomena. In this talk, I will describe these K-moduli spaces for degrees 2, 3, 4, by establishing an isomorphism between the K-moduli spaces and variations of Geometric Invariant Theory compactifications, which generalizes the isomorphisms in the absolute cases established by Odaka--Spotti--Sun and Mabuchi--Mukai. I will also describe the wall-crossing structure for degrees 5,6,7,8,9, that are obtained by computational methods. This is joint work with J. Martinez—Garcia and J. Zhao.