An analysis of concurrent extreme events of continental precipitation and Integrated Water Vapour Transport (IVT) is crucial to our understanding of the role of the major global mechanisms of atmospheric moisture transport, including that of the landfalling Atmospheric Rivers (ARs) in extratropical regions. For this purpose, gridded data on CPC precipitation and ERA-5 IVT at a spatial resolution of 0.5º were used to analyze these concurrent events, covering the period from Winter 1980/1981 to Autumn 2017. For each season, and for each point with more than 400 non-dry days, several copula models were fitted to model the joint distribution function of the two variables. At each of the analysed points, the best copula model was used to estimate the probability of a concurrent extreme. At the same time, within the sample of observed concurrent extremes, the proportion of days with landfalling ARs was calculated for the whole period and for two 15-year sub-periods, one earlier period and one more recent (warmer) period. Three metrics based on copulas were used to analyse carefully the influence of IVT on extreme precipitation in the main regions of occurrence of AR landfall. The results show that the probability of occurrence of concurrent extremes is strongly conditioned by the dynamic component of the IVT, the wind. The occurrence of landfalling ARs accounts for most of the concurrent extreme days of IVT and continental precipitation, with percentages of concurrent extreme days close to 90% in some seasons in almost all the known regions of maximum occurrence of landfalling ARs, and with percentages greater than 75% downwind of AR landfall regions. This coincidence was lower in tropical regions, and in monsoonal areas in particular, with percentages of less than 50%. With a few exceptions, the role of landfalling ARs as drivers of concurrent extremes of IVT and continental precipitation tends to show a decrease in recent (warmer) periods. For almost all the landfalling AR regions with high or very high probabilities of achieving a concurrent extreme, there is a general trend towards a lower influence of IVT on extreme continental precipitation in recent (warmer) periods.