The warm Atlantic Water transported into the Barents Sea plays a crucial role in winter sea ice extent, marine ecosystems, and mid-latitude weather. The North Atlantic Oscillation is known to be an important driver for the Atlantic Water transport variability in the Barents Sea Opening. Here, we find that the dependence of the Barents Sea Opening ocean volume transport variability on the North Atlantic Oscillation is non-stationary. Our results indicate that for the period 1995 to 2005, the link between the North Atlantic Oscillation and the transport variability in the Barents Sea Opening temporarily weakened before an eventual recovery. During this period, synoptic cyclones with unusual trajectories as a consequence of pronounced atmospheric blocking in the North Atlantic sector altered the large-scale and local wind patterns. This temporarily caused a state that the Barents Sea Opening transport variability is largely locally driven instead of being driven by the North Atlantic Oscillation. Our study suggests that an adequate representation of both the North Atlantic Oscillation and cyclone activity is necessary for climate models to better predict future changes in poleward ocean heat transport and Arctic climate.