The transformation of total graph structures has been studied from the algebraic point of view over more than two decades now, and it has motivated the development of the so-called double-pushout and single-pushout approaches to graph transformation. In this article we extend the double-pushout approach to the algebraic transformation of partial many-sorted unary algebras. Such a generalization has been motivated by the need to model the transformation of structures which are richer and more complex than acyclic graphs and hypergraphs. The main result presented in this article is an algebraic characterization of the double-pushout transformation in the categories of all homomorphisms and all closed homomorphisms of unary partial algebras over a given signature, together with a corresponding operational characterization which may serve as a basis for implementation. Moreover, both categories are shown to satisfy the strongest of the HLR (High Level Replacement) conditions with respect to closed monomorphisms. HLR conditions are fundamental to rewriting because they guarantee the satisfaction of many rewriting theorems concerning confluence, parallelism and concurrency.