We present narrowband imagery and long-slit intermediate-resolution spectroscopy of the multiple-shell planetary nebulae NGC 2438 and NGC 5882. A comparative study of the chemical abundances of the main nebulae and their surrounding halos shows a moderate N/O enrichment of the central regions. The oxygen abundances appear constant throughout the nebulae, but there are some indications of a moderate helium enrichment in the NGC 5882 main nebula. This study, along with with data published in the literature for six other multiple-shell planetary nebulae (M1-46, NGC 6543, NGC 6720, NGC6751, NGC 6826, and NGC 7662), provides insight into the chemical enrichment and physical conditions of halos. All the nebulae in this sample have chemical abundances typical of type II planetary nebulae, thus indicating a small chemical enrichment. In addition, we do not find evidence to support a strong chemical enrichment of the main nebulae relative to the halos. The conclusion is that the third dredge-up, in these nebulae, is of small or null effect. Finally, we find a clear evidence that the gas in the halos is ionized by stellar UV radiation leaking through the material of the main nebula.