  We generalize leading-order asymptotics of a form of the heat content of a
submanifold (van den Berg & Gilkey 2015) to the setting of time-dependent
diffusion processes in the limit of vanishing diffusivity. Such diffusion
processes arise naturally when advection-diffusion processes are viewed in
Lagrangian coordinates. We prove that as diffusivity $\varepsilon$ goes to
zero, the diffusive transport out of a material set $S$ under the
time-dependent, mass-preserving advection-diffusion equation with initial
condition given by the characteristic function $\mathds{1}_S$, is
$\sqrt{\varepsilon/\pi} d\overline{A}(\partial S) + o(\sqrt{\varepsilon})$. The
surface measure $d\overline A$ is that of the so-called geometry of mixing, as
introduced in (Karrasch & Keller, 2020). We apply our result to the
characterisation of coherent structures in time-dependent dynamical systems.
