A detailed study of the existence, causal character and multiplicity of geodesics joining two points is carried out for a wide family of non-static Lorentz manifolds (including intermediate Reissner-Nordström, inner Schwarzschild and Generalized Robertson-Walker spacetimes). Results relating causality and connectedness by timelike or lightlike geodesics are obtained, in the spirit of the well-known Avez-Seifert result. The existence of closed spacelike geodesics is also characterized.