  We investigate the impact of transport current on the superconducting order
parameter in superconducting islands in full-shell epitaxial Al-InAs nanowires.
Depending on a device layout, the suppression of superconductivity occurs in
three fundamentally different ways -- by a critical current in the case of
superconducting reservoirs and by a critical voltage or by a critical Joule
power in the case of normal reservoirs. In the latter case, the collapse of the
superconducting state depends on the ratio of the dwell time and the
electron-phonon relaxation time of quasiparticles in the island. For low
resistive and high resistive coupling to the reservoirs, respectively, the
relaxation-free regime and the strong electron-phonon relaxation regime are
realized. Our results shed light on potential shortcomings of finite-bias
transport spectroscopy in floating islands.
