Among confirmed central compact objects (CCOs), only three sources have measured period and period derivatives. We have investigated possible evolutionary paths of these three CCOs in the fallback disc model. The model can account for the individual X-ray luminosities and rotational properties of the sources consistently with their estimated supernova ages. For these sources, reasonable model curves can be obtained with dipole field strengths ~ a few× 109 G on the surface of the star. The model curves indicate that these CCOs were in the spin-up state in the early phase of evolution. The spin-down starts, while accretion is going on, at a time t ~ 103-104 yr depending on the current accretion rate, period, and the magnetic dipole moment of the star. This implies that some of the CCOs with relatively long periods, weak dipole fields, and high X-ray luminosities could be strong candidates to show spin-up behaviour if they indeed evolve with fallback discs.