Exploring the origin of Lyα nebulae ("blobs") at high redshiftrequires measurements of their gas kinematics that are impossible withonly the resonant, optically thick Lyα line. To define gas motionsrelative to the systemic velocity of the blob, the Lyα line mustbe compared with an optically thin line like Hα λ6563,which is not much altered by radiative transfer effects and is moreconcentrated about the galaxies embedded in the nebula's core. We obtainoptical and near-IR (NIR) spectra of the two brightest Lyα blobs(CDFS-LAB01 and CDFS-LAB02) from the Yang et al. sample using theMagellan/Magellan Echellette Spectrograph optical and Very LargeTelescope/SINFONI NIR spectrographs. Both the Lyα and Hαlines confirm that these blobs lie at the survey redshift, z ~ 2.3.Within each blob, we detect several Hα sources, which roughlycorrespond to galaxies seen in Hubble Space Telescope rest-frame UVimages. The Hα detections show that these galaxies have largeinternal velocity dispersions (σ v = 130-190 kms-1) and that, in the one system (LAB01), where we canreliably extract profiles for two Hα sources, their velocitydifference is Δv ~ 440 km s-1. The presence of multiplegalaxies within the blobs, and those galaxies' large velocitydispersions and large relative motion, is consistent with our previousfinding that Lyα blobs inhabit massive dark matter halos that willevolve into those typical of present-day rich clusters and that theembedded galaxies may eventually become brightest cluster galaxies. Todetermine whether the gas near the embedded galaxies is predominantlyinfalling or outflowing, we compare the Lyα and Hα linecenters, finding that Lyα is not offset (ΔvLyα = +0 km s-1) in LAB01 and redshifted byonly +230 km s-1 in LAB02. These offsets are small comparedto those of Lyman break galaxies, which average +450 km s-1and extend to about +700 km s-1. In LAB02, we detect C IIλ1334 and Si II λ1526 absorption lines, whose bluewardshifts of ~200 km s-1 are consistent with the small outflowimplied by the redward shift of Lyα. We test and rule out thesimplest infall models and those outflow models with super/hyperwinds,which require large outflow velocities. Because of the unknown geometryof the gas distribution and the possibility of multiple sources ofLyα emission embedded in the blobs, a larger sample and moresophisticated models are required to test more complex or a wider rangeof infall and outflow scenarios.