We present the detection and follow-up observations of planetary candidates around low-mass stars observed by the K2 mission. Based on light-curve analysis, adaptive-optics imaging, and optical spectroscopy at low and high resolution (including radial velocity measurements), we validate 16 planets around 12 low-mass stars observed during K2 campaigns 5-10. Among the 16 planets, 12 are newly validated, with orbital periods ranging from 0.96 to 33 days. For one of the planets (K2-151b), we present ground-based transit photometry, allowing us to refine the ephemerides. Combining our K2 M-dwarf planets together with the validated or confirmed planets found previously, we investigate the dependence of planet radius R-p on stellar insolation and metallicity [Fe/H]. We confirm that for periods P less than or similar to 2 days, planets with a radius R-p greater than or similar to 2 R-circle plus are less common than planets with a radius between 1-2 R-circle plus. We also see a hint of the "radius valley" between 1.5 and 2 R-circle plus, which has been seen for close-in planets around FGK stars. These features in the radius/period distribution could be attributed to photoevaporation of planetary envelopes by high-energy photons from the host star, as they have for FGK stars. For the M dwarfs, though, the features are not as well defined, and we cannot rule out other explanations such as atmospheric loss from internal planetary heat sources or truncation of the protoplanetary disk. There also appears to be a relation between planet size and metallicity: the few planets larger than about 3 R-circle plus are found around the most metal-rich M dwarfs.