  We study the dynamics of topological defects in continuum theories governed
by a free energy minimization principle, building on our recently developed
framework [Romano J, Mahault B and Golestanian R 2023 J. Stat. Mech.: Theory
Exp. 083211]. We show how the equation of motion of point defects, domain
walls, disclination lines and any other singularity can be understood with one
unifying mathematical framework. For disclination lines, this also allows us to
study the interplay between the internal line tension and the interaction with
other lines. This interplay is non-trivial, allowing defect loops to expand,
instead of contracting, due to external interaction. We also use this framework
to obtain an analytical description of two long-lasting problems in point
defect motion, namely the scale dependence of the defect mobility and the role
of elastic anisotropy in the motion of defects in liquid crystals. For the
former, we show that the effective defect mobility is strongly
problem-dependent, but it can be computed with high accuracy for a pair of
annihilating defects. For the latter, we show that at the first order in
perturbation theory, anisotropy causes a non-radial force, making the
trajectory of annihilating defects deviate from a straight line. At higher
orders, it also induces a correction in the mobility, which becomes
non-isotropic for the $+1/2$ defect. We argue that, due to its generality, our
method can help to shed light on the motion of singularities in many different
systems, including driven and active non-equilibrium theories.
