We report the growth of dilute nitride GaNAs and GaInNAs core-multishell nanowires by plasma-assisted molecular beam epitaxy. Using constituent Ga-induced vapor-liquid-solid growth, these nanowires were grown on Si(111) and silicon on insulator (SOI) substrates. The GaNAs shell nominally contains 0%, 2%, and 3% nitrogen. We also report the growth of GaAs/GaInNAs/GaAs core-multishell nanowires nominally containing 30% In and 2% N. Axial cross-sectional scanning transmission electron microscopy measurements and energydispersive X-ray spectrometry confirm the formation of the core-multishell nanowire structure. We obtained high-quality GaNAs nanowires with nitrogen compositions up to 2%. On the other hand, GaNAs containing 3% nitrogen, and GaInNAs nanowires, show distorted structure; moreover, the optical emissions seem to be related to defects. Further optimisations of the growth conditions will improve these properties, promising future applications in nanoscale optoelectronics.