  We examine the proposal by Grabowska and Kaplan (GK) to use the infinite
gradient flow in the domain-wall formulation of chiral lattice gauge theories.
We consider the case of Abelian theories in detail, for which L\"uscher's exact
gauge-invariant formulation is known, and we relate GK's formulation to
L\"uscher's one. The gradient flow can be formulated for the admissible U(1)
link fields so that it preserves their topological charges. GK's effective
action turns out to be equal to the sum of L\"uscher's gauge-invariant
effective actions for the target Weyl fermions and the mirror "fluffy"
fermions, plus the so-called measureterm integrated along the infinite gradient
flow. The measure-term current is originally a local(analytic) and
gauge-invariant functional of the admissible link field, given as a solution to
the local cohomology problem. However, with the infinite gradient flow, it
gives rise to non-local(non-analytic) vertex functions which are not suppressed
exponentially at large distance. The "fluffy" fermions remain as a source of
non-local contribution, which couple yet to the Wilson-line and magnetic-flux
degrees of freedom of the dynamical link field.
