There is a well-known operation known as cabling for knots, and also for braid groups, where it is a homomorphism $$\beta_k \times \beta_\ell \longrightarrow \beta_{k\ell}$$ given by thickening up the $k$-strand braid and putting one copy of the $\ell$-strand braid inside each of its $k$ tubes. This arises from a map of unordered configuration spaces $C_k(\mathbb{R}^2)\times C_\ell(\mathbb{R}^2) \to C_{k\ell}(\mathbb{R}^2)$, and more generally one has an operation $C_k(M)\times C_\ell(\mathbb{R}^n) \to C_{k\ell}(M)$ for any parallelized manifold $M^n$. More generally again, if one has chosen $a$ linearly independent vector fields on $M$, there is an operation $C_k(M)\times C_\ell(\mathbb{R}^a) \to C_{k\ell}(M)$. In particular, if $M$ admits at least one non-vanishing vector field (either $M$ is non-compact or $\chi(M)=0$) there is a map $C_k(M) \to C_{k\ell}(M)$. This can be thought of as flowing each point of the configuration along the vector field and taking its image at $\ell$ different small time values to obtain the new configuration of $k\ell$ points.

This map has been important in a joint project http://arxiv.org/abs/1406.4916 which I have worked on recently, where we call this the "$\ell$-replication map".

We would be interested in knowing about anywhere in the literature where this has appeared before, as it seems a very natural operation to consider. We would be especially interested in the case of closed manifolds $M$.

One reference which we have come across is the paper http://arxiv.org/abs/math/0701189, which is concerned with this replication map in the case $M=\mathbb{R}^2$.