  We study the stirring process with $N-1$ species on a generic graph
$G=(V,\mathcal{E})$ with reservoirs. The multispecies stirring process
generalizes the symmetric exclusion process, which is recovered in the case
$N=2$. We prove the existence of a dual process defined on an extended graph
$\widetilde{G}=(\widetilde{V},\widetilde{\mathcal{E})}$ which includes
additional extra-sites $\widetilde{V}\setminus V$ where dual particles get
absorbed in the long-time limit. We thus obtain a characterization of the
non-equilibrium steady state of the boundary-driven system in terms of the
absorption probabilities of dual particles. The process is integrable for the
case of the one-dimensional chain with two reservoirs at the boundaries and
with maximally one particle per site. We compute the absorption probabilities
by relying on the underlying ${gl}(N)$ symmetry and the matrix product ansatz.
Thus one gets a closed-formula for (long-ranged) correlations and for the
non-equilibrium stationary measure. Extensions beyond this integrable set-up
are also discussed.
