The paper studies uncertaintly quantification for evolution equations with various time-dependent parameters that
evolve as stochastic processes. Instead of a sensitivity analysis, we measure functionals of the solution, the so-called quantities of interest, by involving scalarizing statistics. The nested distance respects the evolutionary aspect of the problem at hand. Motivated by applications in pedestrian dynamics, we apply these results to drift-diffusion equations in the case when the total mass is-due to boundary or reaction terms-not conserved. We first provide existence and stability for the deterministic problem and then involve uncertainty in the data, which results in Lipschitz continuity with respect to the nested distance. Finally, we present a numerical study illustrating these findings.