[Abridged] With VLT/X-shooter, we obtain optical and NIR spectra of six Ly-alpha blobs at z~2.3. Using three measures --- the velocity offset between the Lya line and the non-resonant [OIII] or H-alpha line (Dv_Lya), the offset of stacked interstellar metal absorption lines, and the spectrally-resolved [OIII] line profile --- we study the kinematics of gas along the line of sight to galaxies within each blob center. These three indicators generally agree in velocity and direction, and are consistent with a simple picture in which the gas is stationary or slowly outflowing at a few hundred km/s from the embedded galaxies. The absence of stronger outflows is not a projection effect: the covering fraction for our sample is limited to <1/8 (13%). The outflow velocities exclude models in which star formation or AGN produce super or hyper winds of up to ~1000km/s. The Dv_Lya offsets here are smaller than typical of LBGs, but similar to those of compact LAEs. The latter suggests that outflow speed cannot be a dominant factor in driving extended Lya emission. For one Lya blob (CDFS-LAB14), whose Lya profile and metal absorption line offsets suggest no significant bulk motion, we use a simple radiative transfer model to make the first column density measurement of gas in an embedded galaxy, finding it consistent with a DLA system. Overall, the absence of clear inflow signatures suggests that the channeling of gravitational cooling radiation into Lya is not significant over the radii probed here. However, one peculiar system (CDFS-LAB10) has a blueshifted Lya component that is not obviously associated with any galaxy, suggesting either displaced gas arising from tidal interactions among blob galaxies or gas flowing into the blob center. The former is expected in these overdense regions, and the latter might signify the predicted but elusive cold gas accretion along filaments.
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