Abstract This paper is about transforming constraint networks to accommodate additional constraints in specific ways. The focus is on two intertwined issues. First, we investigate how partial solutions to an initial network can be preserved from the potential impact of additional constraints. Second, we study how more permissive constraints, which are intended to enlarge the set of solutions, can be accommodated in a constraint network. These two problems are studied in the general case and the light is shed on their relationship. A case study is then investigated where a more permissive additional constraint is taken into account through a form of network relaxation, while some previous partial solutions are preserved at the same time.