We consider a generic quantum conductor coupled to two electric terminals. Phase-coherent transport is generated across the conductor when an electric or thermal gradient is applied to the reservoirs. We investigate heat and charge transport under the influence of inelastic processes, which are modeled using an additional reservoir acting as an ideal volgate and thermal probe. We illustrate our method with an application to a ballistic chaotic cavity. We find within random matrix theory that thermopower fluctuations disappear quickly as the probe mode number increases. Importantly, we find that, upon elimination of the probe, the two-terminal Seebeck coefficient shows an asymmetry when an applied magnetic field reverses its direction. This asymmetry effect is revealed in the presence of incoherent scattering only. Furthermore, the asymmetry effect is of higher order in a Sommerfeld expansion. We believe that our results are relevant to quantify the efficiency of thermoelectric nanodevices in the presence of dephasing.