  We develop the asymptotic analysis as $\epsilon\to 0$ for the natural
gradient flow of the self-dual $U(1)$-Higgs energies
$$E_{\epsilon}(u,\nabla)=\int_M\left(|\nabla
u|^2+\epsilon^2|F_{\nabla}|^2+\frac{(1-|u|^2)^2}{4\epsilon^2}\right)$$ on
Hermitian line bundles over closed manifolds $(M^n,g)$ of dimension $n\ge 3$,
showing that solutions converge in a measure-theoretic sense to codimension-two
mean curvature flows -- i.e., integral $(n-2)$-Brakke flows -- generalizing
results of the last two authors from the stationary case. Given any integral
$(n-2)$-cycle $\Gamma_0$ in $M$, these results can be used together with the
convergence theory developed in previous work of the authors to produce
nontrivial integral Brakke flows starting at $\Gamma_0$ with additional
structure, similar to those produced via Ilmanen's elliptic regularization.
