We examine the sensitivity properties of backward stochastic differential equations and reflected backward stochastic differential equations, which naturally arise in the context of optimal control and optimal stopping problems. Motivated by issues of sensitivity analysis in distributionally robust optimization (DRO) control and optimal stopping problems, we establish explicit formulas for the corresponding sensitivities under drift reference measure uncertainty. Our work is closely related to \citeauthor{bartl2023sensitivity} \cite{bartl2023sensitivity}. In contrast to the existing literature, our analysis is carried out within a general non-Markovian framework.