Context. Tiny molecular clumps are abundant in many H ii regions surrounding newborn stellar clusters. In optical images these so-called globulettes appear as dark patches against the background of bright nebulosity. The majority of the globulettes were found to be of planetary mass in a previous optical investigation, while the largest objects may contain more than half a solar mass. Aims. We aim to clarify the physical nature of globulettes by deriving densities and masses, and to determine their velocities as a function of position over the nebula. This information will provide clues to the question of origins, evolution, and fate of globulettes. The Rosette Nebula is relatively rich in globulettes, and we selected a sample of well-confined objects of different sizes for the present investigation. Methods. Radio observations were made of molecular line emission from 16 globulettes combined with near-infrared (NIR) broadband JHKs and narrow-band Paschen β and H 2 imaging. Ten objects, for which we collected information from several transitions in 12 CO and 13 CO, were modelled using a spherically symmetric model. Results. Practically all globulettes were detected in our CO survey. The observed 12 CO (3-2) and (2-1) line temperatures range from 0.6 K to 6 K, the 13 CO being a third of this. As a rule, the lines are narrow, ∼1.0 km s −1 . The best fit to observed line ratios and intensities was obtained by assuming a model composed of a cool and dense centre and warm and dense surface layer. This model provides estimates of maximum and minimum mass; the average masses range from about 50 to 500 Jupiter masses, which is similar to earlier estimates based on extinction measures. The selected globulettes are dense, n H ∼ 10 4 cm −3 , with very thin layers of fluorescent H 2 emission, showing that the gas is in molecular form just below the surface. The NIR data show that several globulettes are very opaque and contain dense cores. No infrared-excess stars in the fields are associated with globulettes. Internal gas motions are weak, but some larger objects show velocity-shifted components associated with tails. However, most globulettes show no signs of tails or pronounced bright rims in contradiction to current numerical simulations of clumps exposed to intense stellar radiation. Because of the high density encountered already at the surface, the rims become thin, as evidenced by our Pβ images, which also show extended emission that most likely comes from the backside of the globulettes. We conclude that the entire complex of shells, elephant trunks, and globulettes in the northern part of the nebula is expanding with nearly the same velocity of ∼22 km s −1 , and with a very small spread in velocity among the globulettes. We note that the velocities observed for background shells do not fit into a spherically expanding nebular complex. Conclusions. Some globulettes are in the process of detaching from elephant trunks and shells, while other more isolated objects must have detached long ago and are lagging b...