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We consider possibilities to grasp and drag one-dimensional solitons in two-component Bose- Einstein condensates (BECs), under the action of gravity, by tweezers induced by spatially confined spin-orbit (SO) coupling applied to the BEC, with the help of focused laser illumination. Solitons of two types are considered, semi-dipoles and mixed modes. We find critical values of the gravity force, up to which the solitons may be held or transferred by the tweezers. The dependence of the critical force on the magnitude and spatial extension of the localized SO interaction, as well as on the solitons norm and speed (in the transfer regime), are systematically studied by means of numerical methods, and analytically with the help of a quasi-particle approximation for the soliton. In particular, a noteworthy finding is that the critical gravity force increases with the increase of the transfer speed (i.e., moving solitons are more robust than quiescent ones). Nonstationary regimes are addressed too, by considering abrupt application of gravity to solitons created in the weightless setting. In that case, solitons feature damped shuttle motion, provided that the gravity force does not exceed a dynamical critical value, which is smaller than its static counterpart. The results may help to design gravimeters based on ultracold atoms.