Context; Our understanding of the star formation process has traditionally been confined to certain mass or luminosity boundaries because most studies focus only on low-, intermediate- or high-mass star-forming regions. As part of the "Water In Star-forming regions with Herschel" (WISH) key program, water and other important molecules, such as CO and OH, have been observed in 51 embedded young stellar objects (YSOs). The studied sample covers a range of luminosities from <1 to >10^5 L_sol. Aims; We analyse the CO line emission towards a large sample of protostars in terms of both line intensities and profiles. Methods; Herschel-HIFI spectra of the 12CO 10-9, 13CO 10-9 and C18O 5-4, 9-8 and 10-9 lines are analysed for a sample of 51 YSOs. In addition, JCMT spectra of 12CO 3-2 and C18O 3-2 extend this analysis to cooler gas components. Results; All observed CO and isotopologue spectra show a strong linear correlation between the logarithms of the line and bolometric luminosities across six orders of magnitude on both axes. This suggests that the high-J CO lines primarily trace the amount of dense gas associated with YSOs. This relation can be extended to larger (extragalactic) scales. The majority of the detected 12CO line profiles can be decomposed into a broad and a narrow Gaussian component, while the C18O spectra are mainly fitted with a single Gaussian. A broadening of the line profile is also observed from pre-stellar cores to embedded protostars, which is due mostly to non-thermal motions (turbulence/infall). The widths of the broad 12CO 3-2 and 10-9 velocity components correlate with those of the narrow C18O 9-8 profiles, suggesting that the entrained outflowing gas and envelope motions are related independent of the mass of the protostar. These results indicate that physical processes in protostellar envelopes have similar characteristics across the studied luminosity range.Read less <