Abstract The Anosov flows on compact manifolds M satisfy the following property: if p , q are points such that for all positive ϵ there is a trajectory from a point ϵ-close to p to a point ϵ-close to q , then there is a point whose α -limit set is that of p and whose ω -limit set is that of q . Here we give a version of this property for sectional-Anosov flows, namely, vector fields inwardly transverse to the boundary whose maximal invariant set is sectional-hyperbolic. Indeed, if in addition M is three-dimensional and p has non-singular α -limit set, then there is a point whose α -limit set is that of p and whose ω -limit set is either a singularity or that of q .