  We use the {\sc astraeus} framework to investigate how the visibility and
spatial distribution of Lyman-$\alpha$ (Ly$\alpha$) emitters (LAEs) during
reionisation is sensitive to a halo mass-dependent fraction of ionising
radiation escaping from the galactic environment ($f_\mathrm{esc}$) and the
ionisation topology. To this end, we consider the two physically plausible
bracketing scenarios of $f_\mathrm{esc}$ increasing and decreasing with rising
halo mass. We derive the corresponding observed Ly$\alpha$ luminosities of
galaxies for three different analytic Ly$\alpha$ line profiles and associated
Ly$\alpha$ escape fraction ($f_\mathrm{esc}^\mathrm{Ly\alpha}$) models:
importantly, we introduce two novel analytic Ly$\alpha$ line profile models
that describe the surrounding interstellar medium (ISM) as dusty gas clumps.
They are based on parameterising results from radiative transfer simulations,
with one of them relating $f_\mathrm{esc}^\mathrm{Ly\alpha}$ to
$f_\mathrm{esc}$ by assuming the ISM of being interspersed with low-density
tunnels. Our key findings are: (i) for dusty gas clumps, the Ly$\alpha$ line
profile develops from a central to double peak profile as a galaxy's halo mass
increases; (ii) LAEs are galaxies with $M_h\gtrsim10^{10}M_\odot$ located in
overdense and highly ionised regions; (iii) for this reason, the spatial
distribution of LAEs is primarily sensitive to the global ionisation fraction
and only weakly in second-order to the ionisation topology or a halo
mass-dependent $f_\mathrm{esc}$; (iv) furthermore, as the observed Ly$\alpha$
luminosity functions reflect the Ly$\alpha$ emission from more massive
galaxies, there is a degeneracy between the $f_\mathrm{esc}$-dependent
intrinsic Ly$\alpha$ luminosity and the Ly$\alpha$ attenuation by dust in the
ISM if $f_\mathrm{esc}$ does not exceed $\sim50\%$.
