  A well-known property of unordered configuration spaces of points (in an
open, connected manifold) is that their homology stabilises as the number of
points increases. We generalise this result to moduli spaces of submanifolds of
higher dimension, where stability is with respect to the number of components
having a fixed diffeomorphism type and isotopy class. As well as for
unparametrised submanifolds, we prove this also for partially-parametrised
submanifolds -- where a partial parametrisation may be thought of as a
superposition of parametrisations related by a fixed subgroup of the mapping
class group. In a companion paper (arXiv:1807.07558), this is further
generalised to submanifolds equipped with labels in a bundle over the embedding
space, from which we deduce corollaries for the stability of diffeomorphism
groups of manifolds with respect to parametrised connected sum and addition of
singularities.
