  We develop an abstract theory of flows of geometric $H$-structures, i.e.,
flows of tensor fields defining $H$-reductions of the frame bundle, for a
closed and connected subgroup $H\subset SO(n)$, on any connected and oriented
$n$-manifold with sufficient topology to admit such structures. The first part
of the article sets up a unifying theoretical framework for deformations of
$H$-structures, by way of the natural infinitesimal action of
$\mathrm{GL}(n,\mathbb{R})$ on tensors combined with various bundle
decompositions induced by $H$-structures. We compute evolution equations for
the intrinsic torsion under general flows of $H$-structures and, as
applications, we obtain general Bianchi-type identities for $H$-structures,
and, for closed manifolds, a general first variation formula for the
$L^2$-Dirichlet energy functional $\mathcal{E}$ on the space of $H$-structures.
We then specialise the theory to the negative gradient flow of $\mathcal{E}$
over isometric $H$-structures, i.e., their harmonic flow. The core result is an
almost monotonocity formula along the flow for a scale-invariant localised
energy, similar to the classical formulae by Chen-Struwe for the harmonic map
heat flow. This yields an $\varepsilon$-regularity theorem and an energy gap
result for harmonic structures, as well as long-time existence for the flow
under small initial energy, relative to the $L^\infty$-norm of initial torsion,
in the spirit of Chen-Ding. Moreover, below a certain energy level, the absence
of a torsion-free isometric $H$-structure in the initial homotopy class imposes
the formation of finite-time singularities. These seemingly contrasting
statements are illustrated by examples on flat $n$-tori, so long as
$[\mathbb{S}^n,SO(n)/H]$ contains more than one element and the universal cover
of $SO(n)/H$ is a sphere; e.g. when $n=7$ and $H=\rm G_2$, or $n=8$ and $H=\rm
Spin(7)$.
