Context. The Carina nebula hosts a large number of globulettes. An optical study of these tiny molecular clouds shows that the majority are of planetary mass, but there are also those with masses of several tens up to a few hundred Jupiter masses. Aims. We seek to search for, and hopefully detect, molecular line emission from some of the more massive objects; in case of successful detection we aim to map their motion in the Carina nebula complex and derive certain physical properties. Methods. We carried out radio observations of molecular line emission in (CO)-C-12 and (CO)-C-13 (2-1) and (3-2) of 12 globulettes in addition to positions in adjacent shell structures using APEX. Results. All selected objects were detected with radial velocities shifted relative to the emission from related shell structures and background molecular clouds. Globulettes along the western part of an extended dust shell show a small spread in velocity with small velocity shifts relative to the shell. This system of globulettes and shell structures in the foreground of the bright nebulosity surrounding the cluster Trumpler 14 is expanding with a few km s(-1) relative to the cluster. A couple of isolated globulettes in the area move at similar speed. Compared to similar studies of the molecular line emission from globulettes in the Rosette nebula, we find that the integrated line intensity ratios and line widths are very different. The results show that the Carina objects have a different density/temperature structure than those in the Rosette nebula. In comparison the apparent size of the Carina globulettes is smaller, owing to the larger distance, and the corresponding beam filling factors are small. For this reason we were unable to carry out a more detailed modelling of the structure of the Carina objects in the way as performed for the Rosette objects. Conclusions. The Carina globulettes observed are compact and denser than objects of similar mass in the Rosette nebula. The distribution and velocities of these globulettes suggest that they have originated from eroding shells and elephant trunks. Some globulettes in the Trumpler 14 region are quite isolated and located far from any shell structures. These objects move at a similar speed as the globulettes along the shell, suggesting that they once formed from cloud fragments related to the same foreground shell.