We study the topological variant of Rokhlin dimension for topological dynamical systems (X,α,ℤm) in the case where X is assumed to have finite covering dimension. Finite Rokhlin dimension in this sense is a property that implies finite Rokhlin dimension of the induced action on C*-algebraic level, as was discussed in a recent paper by Ilan Hirshberg, Wilhelm Winter and Joachim Zacharias. In particular, it implies (in this context) that the transformation group C*-algebra has finite nuclear dimension. For a single aperiodic homeomorphism, finite Rokhlin dimension follows easily from a recent result by Yonathan Gutman, which in turn uses an important property shown by Elon Lindenstrauss. We show that their methods can be pushed to work in the realm of actions of countably infinite groups. In the particular case of free ℤm-actions on a fixed space X, application of said methods yields a uniform bound of Rokhlin dimension that depends only on m and dim(X).