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JiuBeixin said:

TL;DR: Exploring the relativistic origin of Lorentz force. I'm deriving how a transverse force (from length contraction) and a longitudinal force (from relativistic momentum compensation) combine to stay perpendicular to velocity.

After thinking about this some, I think you have gotten some good advice when you suggested you wanted to learn more about transformation laws.

Maxwell's equations are fully relativistic. If you know how to properly transform charges and currents (usually expressed as charge density and current density), you can use Maxwell's equations to answer your questions. Judging by your questions, you don't currently even know the name of the Lorentz transform - apologies if I got this wrong).

You give the impression of not having any questions about Maxwell's equations (in the most basic form, two Gauss' laws, Faraday's law, and Ampere's law), but rather about the relativistic aspects.

There's a standardized way of describing the source terms. This is in terms of charge density, which would be coulomb/meter^3, and current density, which would be ampere's per meter^2.

The tool you need to know how these quantities transform when switching from a rest frame to a moving frame is called the Lorentz transform, and that's what you need to research.

It'd also be helpful to know the terms "Lorentz boost", which is the way we describe going from a "stationary" frame of reference to a moving frame of reference, and the term "invariant". Invariants are quantites that don't change when you perform a Lorentz boost, I.e. switch from a stationary frame to a moving frame.

The notable invariant of charge and current doesn't have a catchy name, but is given by the quantity

$$c^2 \rho^2 - |J|^2$$

where c is the speed of light, ##\rho## is the charge density in coulomb's/meter^3, and J is the current density in amperes / m^2.

Note that one point of this is that when we boost a neutral current carrying wire in the direction of current flow, it becomes charged. Which you seem to be aware of.

As far as sources go, the results of a "swarm of particles" are well known, but I don't know of many texts that go through the actual work of analyzing a swarm of particles. Usually they just present the results for the continuum limit.