We explore the nature of anomalous conduction of wave packets in disorder-free incommensurate multiwalled carbon nanotubes. The spectrum-averaged diffusion exponent is obtained by calculating the multifractal dimension of the energy spectrum. Depending on the shell chirality, the exponent is found to lie within the range 1/2 <=eta < 1. For large unit-cell mismatch between incommensurate shells, eta approaches the value 1/2 for diffusive motion. The energy-dependent quantum spreading reveals a complex density-of-states-dependent pattern with ballistic, superdiffusive, or diffusive character.