Inspired by a problem of invariance of power series families of probability distributions under binomial thinning, Baron and Wesołowski (Res Math 76:168, 2021) investigated a functional equation being a unification of the Cauchy and the Goła̧b-Schinzel type equations. They determined the solutions of the equation with no or mild regularity assumptions imposed on unknown functions. In this paper we extend their results onto a multi-dimensional case.