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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 incorporates both the TES electrical response, stemming from Josephson phenomena, and the electrothermal effects due to coupling to a thermal bath. Our measurements of a varying dynamic resistance are in excellent agreement with our theory, thereby 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.