Radiative feedbacks associated with natural variability appear to differ significantly from those that are forced, posing a challenge to efforts to constrain global climate sensitivity using the observational record. Here, we present a new conceptual framework for understanding the relationship between forced and unforced radiative feedbacks, and consider how each can be used to constrain climate sensitivity estimates. We show that the time-evolving local radiative feedback following an abrupt forcing can be partitioned into a forced component and an unforced component; while the unforced component, attributable to natural variability, remains roughly constant in time, the forced component decays at a time scale commensurate with a (spatially heterogeneous) characteristic radiative relaxation rate. Using a collection of experiments from the CMIP5 archive, we show that these forced and unforced radiative feedback components can be used to infer lower and upper bounds on the climate sensitivity, respectively. Reanalyses of the instrumental era, on the other hand, only provide estimates of feedbacks due to natural variability, which may be used to infer an upper bound on climate sensitivity. While global feedbacks can also be separated into forced and unforced components, this decomposition is less robust than that computed using local feedbacks, making the latter approach preferable.