R. Schoen and S.-T. Yau [Comment. Math. Helv. 51, 333–341 (1976; Zbl 0361.53040)] proved that a harmonic map from a complete manifold with non-negative Ricci curvature to a compact manifold with non-positive sectional curvature must be constant. The paper under review proves a foliated version of this theorem. This version uses the notions of transversal energy and transversally harmonic maps introduced by J. J. Konderak and R. A. Wolak [Q. J. Math. 54, No. 3, 335–354 (2003; Zbl 1059.53051)] and the main result is as follows: Let \(M\) be a complete foliated manifold of infinite volume such that all leaves are compact, the mean curvature form of the foliation is bounded and coclosed, and the transversal Ricci curvature is non-negative. Let \(M^\prime\) be another foliated manifold for which the transversal sectional curvature is non-positive. Then any transversally harmonic map \(\phi: M\to M^\prime\) of finite transversal energy is transversally constant, i.e., maps all points to the same leaf. Moreover, under the same assumptions without the condition on transversal Ricci curvature for \(M\) they prove that every transversally biharmonic map is transversally harmonic. In particular, their main theorem extends to transversally biharmonic maps.