  Point-like topological defects are singular configurations that occur in a
variety of in and out of equilibrium systems with two-dimensional orientational
order. As they are associated with a nonzero circuitation condition, the
presence of defects induces a long-range perturbation of the orientation
landscape around them. The effective dynamics of defects is thus generally
described in terms of quasi-particles interacting through the orientation field
they produce, whose evolution in the simplest setting is governed by the
diffusion equation. Due to the multivalued nature of the orientation field, its
expression for a defect moving with an arbitrary trajectory cannot be obtained
straightforwardly and is often evaluated in the quasi-static approximation.
Here, we instead propose an approach that allows to derive the exact expression
for the orientation created by multiple moving defects, which we find to depend
on their past trajectories and thus to be nonlocal in time. Performing various
expansions in relevant regimes, we show how improved approximations with
respect to the quasi-static defect solution can be obtained. Moreover, our
results lead to so far unnoticed structures in the orientation field of moving
defects which we discuss in light of existing experimental results.
