  Recently, gamma-ray halos of a few degree extension have been detected around
two middle-aged pulsars, namely, Geminga and PSR B0656+14, by the High Altitude
Water Cherenkov observatory (HAWC). The gamma-ray radiation arise from
relativistic electrons that escape the pulsar wind nebula and diffuse in the
surrounding medium. The diffusion coefficient is found to be significantly
lower than the average value in the Galactic disk. If so, given a typical
transverse velocity of $300-500{\,\rm km /s}$ for a pulsar, its displacement
could be important in shaping the morphology of its gamma-ray halos. Motivated
by this, we study the morphology of pulsar halos considering the proper motion
of pulsar. We define three evolutionary phases of pulsar halo to categorize its
morphological features. The morphology of pulsar halos below 10$\,$TeV is
double peaked or single peaked with an extended tail, which depends on the
electron injection history. Above 10 TeV, the morphology of pulsar halos is
nearly spherical, due to the short cooling timescale ($<50\,$kyr) for tens TeV
electrons. We also quantitatively evaluate the separation between the pulsar
and the center of the gamma-ray halo, as well as the influence of different
assumptions for the pulsar characteristics and the injected electrons. Our
results suggest that the separation between the center of the gamma-ray halo
above 10$\,$TeV and the associated pulsar is usually too small to be observable
by HAWC or LHAASO. Hence, our results provide a useful approach to constrain
the origin of extended sources at very high energies.
