In an effort to detect and understand the origin of large scale motions in the gaseous cosmic web at redshifts 2-4.5 we study the kinematics of the intergalactic medium from velocity shifts between absorption systems common to adjacent pairs of lines of sight to background QSO images. We establish the distribution of velocity shear between the lines of sight for different redshifts and transverse spatial separations up to 300 $h_{70}^{-1}$ physical kpc. Using a simple analytical model of Lyman $\alpha$ clouds as expanding pancakes, and a cosmological $\Lambda$CDM hydro-simulation we find that the observed distribution of velocity shear is consistent with an IGM expanding largely with the Hubble flow. The three dimensional distribution of expansion velocities in the hydro-simulation shows that the underlying velocity field is more complex than just simple expansion: the low density gaseous structures responsible for the Lyman $\alpha$ forest are mostly expanding somewhat faster than the Hubble flow, whereas few structures are undergoing gravitational contraction. We also briefly search for traces of galactic feedback and conclude that high redshift superwinds cannot be dominating the movements of the Ly$\alpha$ forest clouds at the observed epoch.