High-resolution spectra of 123 red giant stars in the globular cluster M13 and 64 red giant stars in M92 were obtained with Hectochelle at the MMT telescope. Emission and line asymmetries in Hα and Ca II K are identified, characterizing motions in the extended atmospheres and seeking differences attributable to metallicity in these clusters and M15. On the red giant branch, emission in Hα generally appears in stars with T eff lsim 4500 K and log L/L sungsim 2.75. Fainter stars showing emission are asymptotic giant branch (AGB) stars or perhaps binary stars. The line-bisector for Hα reveals the onset of chromospheric expansion in stars more luminous than log (L/L sun) ~ 2.5 in all clusters, and this outflow velocity increases with stellar luminosity. However, the coolest giants in the metal-rich M13 show greatly reduced outflow in Hα most probably due to decreased T eff and changing atmospheric structure. The Ca II K3 outflow velocities are larger than shown by Hα at the same luminosity and signal accelerating outflows in the chromospheres. Stars clearly on the AGB show faster chromospheric outflows in Hα than RGB objects. While the Hα velocities on the RGB are similar for all metallicities, the AGB stars in the metal-poor M15 and M92 have higher outflow velocities than in the metal-rich M13. Comparison of these chromospheric line profiles in the paired metal-poor clusters, M15 and M92, shows remarkable similarities in the presence of emission and dynamical signatures, and does not reveal a source of the "second-parameter" effect.