The theory of differential equations driven by measures has lately gained increasing attention, due to the fact that it allows a unified study of classical differential problems, difference equations, impulsive differential equations or dynamic equations on time scales. We focus on measure differential inclusionsformulaThe notation ¹g stands for the Stieltjes measure associated to a nondecreasing left-continuous function g and the multifunction on the right hand side has compact, possibly non-convex values in a real Euclidean space and satisfies bounded variation hypotheses with respect to the Pompeiu excess (and not to the Hausdorff-Pompeiu distance, as in related literature); the key element is a selection principle obtained by V.V. Chistyakov and D. Repovˇs [1]. We present an existence result and we analyze how the solution set changes when we allow perturbations of the function g generating the driving measure.