My point here is that in the boosted frame, there is only an electric field. I am only claiming that the boosted motion should follow the boosted electric field (which it indeed must). I understand your concern that transforming back to the lab frame may not, generally, keep acceleration in...
I have been unable to find a satisfactory explanation of this problem, elsewhere.
Consider an uniform electric field, E, along the y axis. Consider also a uniform magnetic field, B, along the z axis. If we release a particle (charge=q, mass=m) at rest on the origin at time t=0, what will be...