Background: Although there is a substantial body of literature on school leadership and its relationship with student achievement, few studies have examined how change in leadership is related to organizational growth and school improvement. Also less well studied is the influence of contextual conditions on how leadership and organizational processes evolve to constrain/augment school outcomes. Focus of Study: In this study, we use moderated mediation analysis to examine how change in the principal leadership relates to achievement growth, mediated via change in multiple organizational processes -- parent-teacher trust, school climate (measured by school safety), and professional capacity. We further examine how these mediational relationships are moderated by initial school conditions. Research Design: We apply moderated mediation analysis to administrative and survey data of elementary schools from a large urban school district to examine if initial school conditions moderate mediational relationships between school leadership and student outcomes. Conclusions: Our results show that improvements in school leadership are related to student learning gains only through improvements in school climate; this relationship is consistent regardless of whether schools initially had strong or weak leadership, and regardless of whether schools initially had safe or unsafe school climates. [This article was published in "Teachers College Record" (EJ1225416).]