  We study the kinematics of the gaseous cosmic web at high redshift with Lyman
alpha forest absorption in multiple QSO sightlines. Using a simple analytic
model and a cosmological hydrodynamic simulation we constrain the underlying
three-dimensional distribution of velocities from the observed line-of-sight
distribution of velocity shear across the plane of the sky. The distribution is
found to be in good agreement with the intergalactic medium (IGM) undergoing
large scale motions dominated by the Hubble flow. Modeling the Lyman alpha
clouds analytically and with a hydrodynamics simulation, the average expansion
velocity of the gaseous structures causing the Lyman alpha forest in the lower
redshift (z = 2) sample appears about 20 percent lower than the local Hubble
expansion velocity. We interpret this as tentative evidence for some clouds
undergoing gravitational collapse. However, the distribution of velocities is
highly skewed, and the majority of clouds at redshifts from 2 to 3.8 expand
typically about 5 - 20 percent faster than the Hubble flow. This behavior is
explained if most absorbers in the column density range typically detectable
are expanding filaments that stretch and drain into more massive nodes. We find
no evidence for the observed distribution of velocity shear being significantly
influenced by processes other than Hubble expansion and gravitational
instability, like galactic winds. To avoid overly disturbing the IGM, winds may
be old and/or limp by the time we observe them in the Lyman alpha forest, or
they may occupy only an insignificant volume fraction of the IGM. (abridged)
