On harmonic morphisms projecting harmonic functions to harmonic functions

For Riemannian manifolds $M$ and $N$, admitting a submersive harmonic morphism $\phi$ with compact fibres, we introduce the vertical and horizontal components of a real-valued function $f$ on $V\subset M$. By comparing the Laplacians on $M$ and $N$, we determine conditions under which a harmonic function on $V=\phi^{-1}(U)\subset M$ projects down, via its horizontal component, to a a harmonic function on $U\subset N$.