Carbon nanotube field-effect transistors operate over a wide range of electron or hole density, controlled by the gate voltage. Here we calculate the mobility in semiconducting nanotubes as a function of carrier density and electric field, for different tube diameters and temperature. The low-field mobility is a non-monotonic function of carrier density, and varies by as much as a factor of 4 at room temperature. At low density, with increasing field the drift velocity reaches a maximum and then exhibits negative differential mobility, due to the non-parabolicity of the bandstructure. At a critical density $\rho_c\sim$ 0.35-0.5 electrons/nm, the drift velocity saturates at around one third of the Fermi velocity. Above $\rho_c$, the velocity increases with field strength with no apparent saturation.