The Great Plains Low-Level Jet (GPLLJ) system consists of very strong winds in the lower troposphere that transport a huge amount of moisture from the Gulf of Mexico to the American Great Plains. This paper aims to study the extremes of the Transported Moisture (TM) from the GPLLJ source region to the jet domain; and, for low and high TM, to analyze the extremal dependence between the upper tail of the precipitation in the GPLLJ sink region and the lower tail of the tropospheric stability in that region, which is known as tropospheric instability. The declustered extremes of TM were analyzed using Peaks Over Threshold (POT). A non-stationary Exponential model was fitted to the cluster maxima. Estimated return levels show that the extremes of TM are expected to decrease in the future. This is meteorologically congruent with the known displacement of the western edge of the North Atlantic Subtropical High, which controls atmospheric circulation in the North Atlantic, and to a higher scale with the change of phase from negative to positive of the Atlantic Multidecadal Oscillation. Bilogistic and Logistic models were fitted to the extremes of (tropospheric instability, precipitation) for low and high TM, respectively. The extremal dependence between tropospheric instability and precipitation proves to be stronger in the case of high TM. This confirms that dynamical instability is the most important parameter for achieving high values of precipitation once there is a mechanism that allows the continuous supply of large amounts of moisture, such as the derived from a low-level jet system.