We present a kinematic analysis of the atomic and molecular gas in the Ursa Major molecular clouds. The analysis is based on a new CO survey of the complex made with linear resolution of 0.05 pc and existing H I observations. The clouds lie in projection on an expanding shell of material known as the North Celestial Pole loop. The molecular structure of the complex is dominated by several long (>5 pc) filaments, some of which are both extremely straight and extremely narrow (<0.2 pc across). These filaments are enclosed in a sheath of neutral atomic hydrogen that has a kinematic signature distinct from the surrounding atomic gas. The tips of the filaments are regions of enhanced IRAS 12 and 25 μm emission. We find an offset of up to 4 km s-1 between the centroid velocities of the CO and H I, a large-scale velocity gradient in the gas of ~0.3 km s-1 pc-1, and a similar large-scale gradient in the H I line width. The CO velocity field follows a trend similar to the H I but is much less organized. A weak line width gradient in the CO may also be present. We present a model in which the clouds lie near the surface of the expanding bubble and, after having interacted with the bubble's wind, are now slowly sliding down the bubble toward the Galactic plane. This model adequately accounts for the atomic-molecular velocity offset, the velocity gradients, the line width gradient, and the IRAS colors.