In this paper, we obtain a sufficient condition for pluriharmonic mappings on the Euclidean unit ball Bn to be univalent, sense-preserving, quasiconformal and bi-Lipschitz diffeomorphisms on Bn and to have linearly connected images. Also, we give a sufficient condition for pluriharmonic mappings on Bn to have quasiconformal extensions to Cn. Next, we generalize the harmonic Schwarz lemma to pluriharmonic mappings of the unit ball BX of a complex Banach space X into the unit ball Bn of Cn with respect to an arbitrary norm. Further, we obtain a generalization of the harmonic Schwarz–Pick lemma to the case of pluriharmonic mappings of the homogeneous unit ball BX of a complex Banach space X into the unit ball Bn. We also obtain a version of the holomorphic Schwarz–Pick lemma for the Jacobian determinant on the Euclidean unit ball Bn to the case of pluriharmonic mappings of the homogeneous unit ball BX into Bn, in the case that BX is an open subset of Cn. Finally, we obtain the Landau and the Bloch theorems for pluriharmonic or holomorphic mappings on finite dimensional homogeneous unit balls.