A recurrent issue in applications of extreme value analysis is the existence of time-varying structural dependencies. For instance, operational loss severity distribution is often concerned with changes in regulations, business cycles or financial crises whereas income distribution is affected by major political events. These changes, however, can affect progressively rather than abruptly the dependence structure with potential predictors. To help accounting for this empirical feature, we introduce a smooth-transition Generalized Pareto (GP) regression model. In this time-varying regression model, the parameters of the GP distribution are related to explanatory variables through a regression function, which depends itself on time. We illustrate the good finite sample properties of this model with a simulation study. Later on, we study the monthly severity distribution of operational losses at UniCredit over the period 2005-2014 and across three event types. We show that past levels of the VIX are good indicators of time-varying dependencies. We also consider a second application, where we study the income distribution of very rich people in Germany between 1984 and 2013, using the r-largest value approach. We show changes in the dependence structure after the Reunification.