"Swimming" versus "swinging" in spacetime

Phys.Rev. D73 (2006) 024020 Wisdom has recently unveiled a new relativistic effect, called ``spacetime swimming'', where quasi-rigid free bodies in curved spacetimes can "speed up", "slow down" or "deviate" their falls by performing "local" cyclic shape deformations. We show here that for fast enough cycles this effect dominates over a non-relativistic related one, named here ``space swinging'', where the fall is altered through "nonlocal" cyclic deformations in Newtonian gravitational fields. We expect, therefore, to clarify the distinction between both effects leaving no room to controversy. Moreover, the leading contribution to the swimming effect predicted by Wisdom is enriched with a higher order term and the whole result is generalized to be applicable in cases where the tripod is in large red-shift regions.