In the rest frame of a many-body system, used in the calculation of its static and scattering properties, the center of mass of a two-body subsystem is allowed to drift. We show, in a model-independent way, that drift corrections to the nucleon-nucleon potential are relatively large and arise from both one- and two-pion exchange processes. As far as chiral symmetry is concerned, corrections to these processes begin, respectively, at O(q(2)) and O(q(4)). The two-pion exchange interaction also yields a new spin structure, which promotes the presence of P waves in trinuclei and is associated with profile functions that do not coincide with either central or spin-orbit ones. In principle, the new spin terms should be smaller than the O(q(3)) spin-orbit components. However, in the isospin-even channel, a large contribution defies this expectation and gives rise to the prediction of important drift effects.