We theoretically predict and experimentally observe the onset of weak-link
physics in the dynamical response of transition edge sensors (TES). We develop
a theoretical framework based on a Fokker-Planck description that unifies the
TES electrical response, stemming from Josephson phenomena, with electrothermal
effects due to coupling to a thermal bath. Our measurements of a varying
dynamic resistance are in excellent agreement with our theory, thereby ruling
out predictions based on a two-fluid model and establishing weak-link phenomena
as the main mechanism underlying the operation of TES. Furthermore, our
description enables the calculation of power spectral densities, paving the way
for a more thorough investigation of the unexplained "excess noise" in long
diffusive junctions and TES reported in recent experiments