  We present new radial velocity (RV) measurements for 11 candidate young
very-low-mass stars and brown dwarfs, with spectral types from M7 to L7.
Candidate young objects were identified by features indicative of low surface
gravity in their optical and/or near-infrared spectra. RV measurements are
derived from high resolution (R=$\lambda$/$\Delta\lambda$=20,000) $J$ band
spectra taken with NIRSPEC at the Keck Observatory. We combine RVs with proper
motions and trigonometric distances to calculate three-dimensional space
positions and motions and to evaluate membership probabilities for nearby young
moving groups (NYMGs). We propose 2MASS J00452143+1634446 (L2$\beta$,
$J$=13.06) as an RV standard given the precision and stability of measurements
from three different studies. We test the precision and accuracy of our RV
measurements as a function of spectral type of the comparison object, finding
that RV results are essentially indistinguishable even with differences of
$\pm$5 spectral subtypes. We also investigate the strengths of
gravity-sensitive K~{\sc i} lines at 1.24--1.25 $\mu$m and evaluate their
consistency with other age indicators. We confirm or re-confirm four brown
dwarf members of NYMGs -- 2MASS J00452143+1634446, WISE J00470038+6803543,
2MASS J01174748$-$3403258, and 2MASS J19355595$-$2846343 -- and their previous
age estimates. We identify one new brown dwarf member of the Carina-Near moving
group, 2MASS J21543454$-$1055308. The remaining objects do not appear to be
members of any known NYMGs, despite their spectral signatures of youth. These
results add to the growing number of very-low-mass objects exhibiting
signatures of youth that lack likely membership in a known NYMG, thereby
compounding the mystery regarding local, low-density star formation.
