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=$lambda$/$Deltalambda$=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.
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