We address the issue of Ontology-Based Data Access, with ontologies represented in the framework of existential rules, also known as Datalog+/-.A well-known approach involves rewriting the query using ontological knowledge. We focus here on the basic rewriting technique which consists of rewriting a conjunctive query (CQ) into a union of CQs. We assume that the set of rules is a finite unification set, i.e., for any CQ, there exists a finite sound and complete rewriting of this CQ with the rules. First, we study a generic breadth-first rewriting algorithm, which takes as input any rewriting operator. We define properties of the rewriting operator that ensure the correctness and the termination of this algorithm. Second, we study some operators with respect to the exhibited properties. All these operators have in common to be based on so-called piece-unifiers but they lead to different explorations of the rewriting space. Finally, an experimental comparison of these operators within an implementation of the generic breadth-first rewriting algorithm is presented.