Logical relations built on top of an operational semantics are one of the most successful proof methods in programming language semantics. In recent years, more and more expressive notions of operationally-based logical relations have been designed and applied to specific families of languages. However, a unifying abstract framework for operationally-based logical relations is still missing. We show how fibrations can provide a uniform treatment of operational logical relations, using as reference example a $\lambda$-calculus with generic effects endowed with a novel, abstract operational semantics defined on a large class of categories. Moreover, this abstract perspective allows us to give a solid mathematical ground also to differential logical relations --- a recently introduced notion of higher-order distance between programs --- both pure and effectful, bringing them back to a common picture with traditional ones.