**Abstract**

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.