We study how the sizes and radial profiles of galaxies vary with wavelength, by fitting Sérsic functions simultaneously to imaging in nine optical and near-infrared bands. To quantify the wavelength dependence of effective radius we use the ratio, RR, of measurements in two rest-frame bands. The dependence of Sérsic index on wavelength, NN, is computed correspondingly. Vulcani et al. have demonstrated that different galaxy populations present sharply contrasting behaviour in terms of RR and NN. Here we study the luminosity dependence of this result. We find that at higher luminosities, early-type galaxies display a more substantial decrease in effective radius with wavelength, whereas late types present a more pronounced increase in Sérsic index. The structural contrast between types thus increases with luminosity. By considering samples at different redshifts, we demonstrate that lower data quality reduces the apparent difference between the main galaxy populations. However, our conclusions remain robust to this effect. We show that accounting for different redshift and luminosity selections partly reconciles the size variation measured by Vulcani et al. with the weaker trends found by other recent studies. Dividing galaxies by visual morphology confirms the behaviour inferred using morphological proxies, although the sample size is greatly reduced. Finally, we demonstrate that varying dust opacity and disc inclination can account for features of the joint distribution of RR and NN for late-type galaxies. However, dust does not appear to explain the highest values of RR and NN. The bulge–disc nature of galaxies must also contribute to the wavelength dependence of their structure.