Understanding the dynamics and evolution of climate change and associated uncertainties is key for designing robust policy actions. Computer models are key tools in this scientific effort, which have now reached a high level of sophistication and complexity. Model auditing is needed in order to better understand their results, and to deal with the fact that such models are increasingly opaque with respect to their inner workings. Current techniques such as Global Sensitivity Analysis (GSA) are limited to dealing either with multivariate outputs, stochastic ones, or finite-change inputs. This limits their applicability to time-varying variables such as future pathways of greenhouse gases. To provide additional semantics in the analysis of a model ensemble, we provide an extension of GSA methodologies tackling the case of stochastic functional outputs with finite change inputs. To deal with finite change inputs and functional outputs, we propose an extension of currently available GSA methodologies while we deal with the stochastic part by introducing a novel, domain-selective inferential technique for sensitivity indices. Our method is explored via a simulation study that shows its robustness and efficacy in detecting sensitivity patterns. We apply it to real world data, where its capabilities can provide to practitioners and policymakers additional information about the time dynamics of sensitivity patterns, as well as information about robustness.