The neutron–proton effective mass splitting in neutron-rich nucleonic matter reflects the spacetime nonlocality of the isovector nuclear interaction. It affects the neutron/proton ratio during the earlier evolution of the Universe, cooling of proto-neutron stars, structure of rare isotopes and dynamics of heavy-ion collisions. While there is still no consensus on whether the neutron–proton effective mass splitting is negative, zero or positive and how it depends on the density as well as the isospin-asymmetry of the medium, significant progress has been made in recent years in addressing these issues. There are different kinds of nucleon effective masses. In this mini-review, we focus on the total effective masses often used in the non-relativistic description of nuclear dynamics. We first recall the connections among the neutron–proton effective mass splitting, the momentum dependence of the isovector potential and the density dependence of the symmetry energy. We then make a few observations about the progress in calculating the neutron–proton effective mass splitting using various nuclear many-body theories and its effects on the isospin-dependence of in-medium nucleon–nucleon cross-sections. Perhaps, our most reliable knowledge so far about the neutron–proton effective mass splitting at saturation density of nuclear matter comes from optical model analyses of huge sets of nucleon–nucleus scattering data accumulated over the last five decades. The momentum dependence of the symmetry potential from these analyses provide a useful boundary condition at saturation density for calibrating nuclear many-body calculations. Several observables in heavy-ion collisions have been identified as sensitive probes of the neutron–proton effective mass splitting in dense neutron-rich matter based on transport model simulations. We review these observables and comment on the latest experimental findings.