According to the current paradigm, galaxies initially form as disc galaxies at the centres of their own dark matter haloes. During their subsequent evolution, they may undergo a transformation to a red, early-type galaxy, thus giving rise to the build-up of the red sequence. Two important, outstanding questions are (i) which transformation mechanisms are most important and (ii) in what environment do they occur. In this paper, we study the impact of transformation mechanisms that operate only on satellite galaxies, such as strangulation, ram-pressure stripping and galaxy harassment. Using a large galaxy group catalogue constructed from the Sloan Digital Sky Survey, we compare the colours and concentrations of satellites galaxies to those of central galaxies of the same stellar mass, adopting the hypothesis that the latter are the progenitors of the former. On average, satellite galaxies are redder and more concentrated than central galaxies of the same stellar mass, indicating that satellite-specific transformation processes do indeed operate. Central-satellite pairs that are matched in both stellar mass and colour, however, show no average concentration difference, indicating that the transformation mechanisms operating on satellites affect colour more than morphology. We also find that the colour and concentration differences of matched central-satellite pairs are completely independent of the mass of the host halo (not to be confused with the subhalo) of the satellite galaxy, indicating that satellite-specific transformation mechanisms are equally efficient in host haloes of all masses. This strongly rules against mechanisms that are thought to operate only in very massive haloes, such as ram-pressure stripping or harassment. Instead, we argue that strangulation is the main transformation mechanism for satellite galaxies. Finally, we determine the relative importance of satellite quenching for the build-up of the red sequence. We find that roughly 70 per cent of red-sequence satellite galaxies with M*∼ 109h−2M⊙ had their star formation quenched as satellites. This drops rapidly with increasing stellar mass, reaching virtually zero at M*∼ 1011h−2M⊙. Therefore, a very significant fraction of red satellite galaxies were already quenched before they became a satellite.