  It is widely believed that many-body localisation in one dimension is fragile
and can be easily destroyed by thermal inclusions, however there are still many
open questions regarding the stability of the localised phase and under what
conditions it breaks down. Here I construct models with dilute disorder, which
interpolate between translationally invariant and fully random models, in order
to study the breakdown of localisation. This opens up the possibility to
controllably increase the density of thermal regions and examine the breakdown
of localisation as this density is increased. At strong disorder, the numerical
results are consistent with commonly-used diagnostics for localisation even
when the concentration of thermalising regions is high. At moderate disorder, I
present evidence for slow dynamics and sub-diffusive transport across a large
region of the phase diagram, suggestive of a `bad metal' phase. This suggests
that dilute disorder may be a useful effective model for studying Griffiths
effects in many-body localisation, and perhaps also in a wider class of
disordered systems.
