  We prove the existence of a canonical `higher Kolyvagin derivative'
homomorphism between the modules of higher rank Euler systems and higher rank
Kolyvagin systems, as has been conjectured to exist by Mazur and Rubin. This
homomorphism exists in the setting of $p$-adic representations that are free
with respect to the action of a Gorenstein order $\mathcal{R}$ and, in
particular, implies that higher rank Euler systems control the
$\mathcal{R}$-module structures of Selmer modules attached to the
representation. We give a first application of this theory by considering the
(conjectural) Euler system of Rubin-Stark elements.
