Fairness-related assumptions about what constitutes appropriate NLG system behaviors range from invariance, where systems are expected to respond identically to social groups, to adaptation, where responses should instead vary across them. We design and conduct five case studies, in which we perturb different types of identity-related language features (names, roles, locations, dialect, and style) in NLG system inputs to illuminate tensions around invariance and adaptation. We outline people's expectations of system behaviors, and surface potential caveats of these two contrasting yet commonly-held assumptions. We find that motivations for adaptation include social norms, cultural differences, feature-specific information, and accommodation; motivations for invariance include perspectives that favor prescriptivism, view adaptation as unnecessary or too difficult for NLG systems to do appropriately, and are wary of false assumptions. Our findings highlight open challenges around defining what constitutes fair NLG system behavior.