We investigate three-sided matching problems where three kinds of agents, men, women, and cats are matched. Without any restrictions on preferences of agents, a stable matching does not necessarily exist for a three-sided matching problem. However, Danilov  has proved the existence of a stable matching for any three-sided matching problem if preference domains for men and women are restricted in a certain way. In the present paper, we show that, starting from an arbitrary unstable matching, there exists a finite sequence of successive blockings leading to some stable matching for a three-sided matching problem in Danilov’s model, as Roth and Vande Vate  have proved for two-sided matching problems. The result implies that a decentralized process of successive blockings by randomly chosen blocking agents will converge to a stable matching.