I will try to repeat your argument more precisely, to make sure I understand: you are saying that if you transform coordinates linearly, you take linear trajectories to linear trajectories, but if you transform coordinates nonlinearly, you take linear trajectories to curved trajectories, and this makes you think that an incoming pair of particles coming in at a straight lines will be transformed into particles which move along curvilinear trajectories, and therefore feel an instant force at a distance.
This is not a problem at all, because the conformal invariance is not an invariance of particle trajectories. A particle trajectory is constructed as a state using an operator which is necessarily smeared over a large region of space, where the function is periodic along planes corresponding to wavefronts to the extent it is extended in time (you are free to create a particle with a localized pulse, but I am assuming you are doing the creation gradually in time), and when you do a conformal transformation, you transform the smearing function to produce a nonlocal creation process which produces a curved trajectory.
The invariance of the correlation functions under conformal transformation just doesn't imply an invariance of particle trajectories under conformal transformations.
This post imported from StackExchange Physics at 2014-04-24 02:33 (UCT), posted by SE-user Ron Maimon