We consider the class of forward-closed theories defined as the equational theories generated by forward-closed convergent rewrite systems. In this particular class of syntactic theories, the unification problem can be solved thanks to a mutation-based unification algorithm. This is shown by reusing an approach based on Basic Syntactic Mutation originally developed for equational theories saturated by paramodulation. Then this mutation-based algorithm is applied in a new combination algorithm that solves the unification problem in some non-disjoint extensions of forward-closed theories.