We discuss the geometry of partially decoupling (submersive) second-order equations in general and illustrate the theory with an application to the case of Lagrangian systems of mechanical type: it is shown that submersiveness implies decoupling into separate subsystems in that case, unless non-conservative forces are added to the system. The main purpose of the paper is to explain the geometric structures underlying so called driven cofactor systems, which constitute a special class of non-conservative Lagrangian systems. In doing so, we generalize the original set-up of driven cofactor systems (Lundmark and Rauch-Wojciechowski 2002 J. Math. Phys. 43 6166) from a Euclidean space to an arbitrary Riemannian one.