  We formulate the gradient Dirichlet flow of $Sp(2)Sp(1)$-structures on
$8$-manifolds, as the first systematic study of a geometric quaternion-K\"ahler
(QK) flow. Its critical condition of \emph{harmonicity} is especially relevant
in the QK setting, since torsion-free structures are often topologically
obstructed. We show that the conformally parallel property implies harmonicity,
extending a result of Grigorian in the $G_2$ case. We also draw several
comparisons with $Spin(7)$-structures. Analysing the QK harmonic flow, we prove
an almost-monotonicity formula, which implies to long-time existence under
small initial energy, via $\epsilon$-regularity. We set up a theory of harmonic
QK solitons, constructing a non-trivial steady example. We produce explicit
long-time solutions: one, converging to a torsion-free limit on the hyperbolic
plane; and another, converging to a limit which is harmonic but not
torsion-free, on the manifold $SU(3)$. We also study compactness and the
formation of singularities.
