  In this paper, we continue to study the fractional harmonic gradient flow on
$S^{n-1}$ taking values in a general closed manifold $N \subset \mathbb{R}^n$,
addressing global existence and uniqueness of solutions of energy class with
sufficiently small energy, adding to the existing body of knowledge pertaining
to the half-harmonic gradient flow and expanding upon our previous work in
[34]. We extend the techniques by Struwe in [30] and Rivi\`ere in [22] to the
non-local framework analogous to [34] to derive uniqueness, employ commutator
estimates as in [8] for regularity and follow [30] for a general existence
result.
