The Chester-Thellung relation between transport coefficients of a degenerate Fermi gas is used to extend recent results on the long-range character of the conductivity tensor to thermoelectric and heat conductivity tensors. Consequences are discussed for combined voltage and temperature measurements in multilead devices. Among our predictions is that the temperature and temperature-induced voltage drops determined via a four-probe measurement should fluctuate dramatically, both in sign and magnitude, as a function of the chemical potential and external magnetic field. If more than four probes are used, any two measured temperature and temperature-induced voltage drops will be uncorrelated, both in sign and magnitude, at fixed values of chemical potential and magnetic field. We propose two representative experiments, which involve the application of a temperature rather than a voltage difference between the two end probes. (i) A grounded sample which is thermally isolated except at the probes. Here, large temperature fluctuations will be observed. (ii) A sample in good thermal contact with the substrate, but electrically isolated except at the probes. Here, induced voltage will show large fluctuations.