The term Weak Stability Boundary (WSB) is related to a region of stable motion around the second primary of a circular restricted three-body problem (CR3BP). Previous work on this subject has shown that, at a given energy level, the boundaries of such region are provided by the stable manifolds of central objects of the L1 and L2 libration points, i.e., the two planar Lyapunov orbits (PLOs). This offers a natural dynamical channel between the Earth's vicinity and the Sun-Earth libration points L1 and L2. Furthermore, it has been shown (and successfully employed to design low-energy spacecraft lunar transfers) that the Sun-Earth L2 central unstable manifolds can be linked, through an heteroclinic connection, to the central stable manifolds of the L2 point in the Earth-Moon three-body problem. The aim of the present study is to clarify the relationship between the low-energy Earth-to-Moon transfers (LETs) and the dynamics of the phase space points that populate the WSB region around the Earth. The present work develops through an extensive and systematic exploration of the trajectories connecting planar Lyapunov orbits corresponding to all the possible combinations of two libration points in the Sun-Earth and Earth-Moon CR3BPs, kinematically coupled. The results of such exploration give us a deeper and more 2 complete understanding of the dynamics and properties of such connections and constitute the basis for the next stage of the investigation, that is the study of the structure of the WSB around the Earth, its dynamical relationship first with the Sun- Earth libration points L1 and L2 and then with the Earth-Moon ones, in the bicircular four-body model. This investigation is part of a research work that will be the subject of a subsequent, more extended publication.