In the general context of the theory of institutions, several notions of parchment and parchment morphism have been proposed as the adequate setting for combining logics. However, so far, they seem to lack one of the main advantages of the combination mechanism known as fibring: general results of transference of important logical properties from the logics being combined to the resulting fibred logic. In previous work, we have proposed a setting for bringing fibring to the realm of institutions through the novel notion of c-parchment, and managed to prove several soundness and completeness preservation results for propositional-based logics. Herein, we extend these results to c-parchments also encompassing logics with variables, terms and quantifiers. As a working example we look into modal first-order logic as a fibring of modal propositional logic and first-order logic, and compare the results with several of the alternative semantics proposed in the literature.