We study \({({\mathcal{F}}, {\mathcal{G}})}\)-harmonic maps between foliated Riemannian manifolds \({(M, {\mathcal{F}}, g)}\) and \({(N, {\mathcal{G}}, h)}\) i.e. smooth critical points ϕ : M → N of the functional \({E_T (\phi ) = \frac{1}{2} \int_M \| d_T \phi \|^2 \,d \, v_g}\) with respect to variations through foliated maps. In particular we study \({({\mathcal{F}}, {\mathcal{G}})}\)-harmonic morphisms i.e. smooth foliated maps preserving the basic Laplace equation Δ B u = 0. We show that CR maps of compact Sasakian manifolds preserving the Reeb flows are weakly stable \({({\mathcal{F}}, {\mathcal{G}})}\)-harmonic maps. We study \({({\mathcal{F}}, {\mathcal{G}}_0 )}\)-harmonic maps into spheres and give foliated analogs to Solomon’s (cf., J Differ Geom 21:151–162, 1985) results.