One of the first examples of problems involving Maxwell's electromagnetism used in order to introducing special relativity is the problem of a charged particle $q$ in a stationary magnetic field, let's say $\vec{B} = B_0 \vec{e}_z$. If we are in a rest frame relatively to the particle, the charge is motionless and so there is no Lorentz force applied and the particle remain still.
If the observer is on a inertial system moving with a relative velocity $\vec{v}$ in relation to the first system, the particle would have a velocity of $-\vec{v}$ and should move in response to the Lorentz force $\vec{F}=\vec{v} \times \vec{B}$. As far as I know, this paradox is solved using Lorentz transformation between the two systems.
Hence there is my doubt: when speaking of kinematics (for example lengths contraction), it is clear that at low velocity the relativistic effect are neglegible and this is even more clear looking how Lorentz transformation tend to Galilean ones for $v<<c$. However, this classical limit seems not to be appliable in the electrodynamics case because even at very low speed the Lorentz force make a great difference between the two scenarios.
If the Lorentz transformation have the Galilean ones as a limit for low velocities, why this is not appliable in electrodynamics?