  Recent Pulsar Timing Array (PTA) collaborations show strong evidence for a
stochastic gravitational wave background (SGWB) with the characteristic
Hellings-Downs inter-pulsar correlations. The signal may stem from supermassive
black hole binary mergers, or early universe phenomena. The former is expected
to be strongly anisotropic while primordial backgrounds are likely to be
predominantly isotropic with small fluctuations. In case the observed SGWB is
of cosmological origin, our relative motion with respect to the SGWB rest frame
is a guaranteed source of anisotropy, leading to $\textit{O}(10^{-3})$ energy
density fluctuations of the SGWB. For such cosmological SGWB, kinematic
anisotropies are likely to be larger than the intrinsic anisotropies, akin to
the cosmic microwave background (CMB) dipole anisotropy. We assess the
sensitivity of current PTA data to the kinematic dipole anisotropy, and we also
forecast at what extent the magnitude and direction of the kinematic dipole can
be measured in the future with an SKA-like experiment. We also discuss how the
spectral shape of the SGWB and the location of the pulsars to monitor affect
the prospects of detecting the kinematic dipole with PTA. In the future, a
detection of this anisotropy may even help resolve the discrepancy in the
magnitude of the kinematic dipole as measured by CMB and large-scale structure
observations.
