  For a manifold $W$ and an $E_d$-algebra $A$, the factorisation homology
$\int_W A$ can be seen as a generalisation of the classical configuration space
of labelled particles in $W$. It carries an action by the diffeomorphism group
$\mathrm{Diff}_\partial(W)$, and for the generalised surfaces $W_{g,1}:=(\#^g
S^n\times S^n)\setminus\mathring D{}^{2n}$, we have stabilisation maps among
the quotients $\int_{W_{g,1}} A\,/\!/\,\mathrm{Diff}_\partial(W_{g,1})$ which
increase the genus $g$. In the case where a highly-connected tangential
structure $\theta$ is taken into account, we describe its stable homology in
terms of the iterated bar construction $\mathrm{B}^{2n}A$ and a tangential Thom
spectrum $\mathrm{MT}\theta$. We also consider the question of homological
stability.
