The question is that the transformation of these magnitudes is always local and it is not obvious for me how it could be measured even in a given system of coordinates for an external observer of the material which contains them.
If I may translate your question relative to the transformation of the electric and magnetic oscillations into terms of harmonic magnitudes - specifically, “how it could be measured in a given system of coordinates” - which refers to the quantum frequencies of the material and the natural units of measure.
The answer which follows would be based on intervals and measures defined by quantum frequencies of the electronic and atomic configuration, which relate directly to the harmonic architecture of the Prime Root Matrix - which governs the prime root frequencies of all electrons, atoms and molecules.
Each atom has a set of prime roots that represent the orbital functions of the electrons - and the prime roots for all of the elements are part of a synchronous numerical system that governs all vibronic coupling - for instance, super-conduction and the combinatorial frequencies of base pairs of electrons, which function in accordance to the decimal system and base paired prime roots.
The resonant intervals of electromagnetic fluctuations are quantum frequencies commensurate with the Quantum Gravity Matrix - measured in sequential integer units of time - in seconds, which numerically correspond to the periodicity of the Prime Root Matrix and the Universal Number System which give rise to the harmonic and rhythmic wave mechanics of three dimensional action.
The sequence of Prime Roots naturally function in cohesive unison when properly ordered, according to the Universal harmonic number system - which govern the electromagnetic properties of the Periodic Table of Elements - and reveals the quantum frequencies of those elements discovered so far - and quantum frequencies of those elements yet to be discovered.
Therefore, the key to all quantum frequencies of the energy spectrum - and the transformation of electrical and magnetic forces - lies in the domain of how the Prime Root Matrix is empirically orchestrated as a harmonic number system - which has historically been an enigma...until now.
Thank you for your answer, but I cannot follow it and to understand about what are speaking on transformations. Perhaps it was my mistake when I have put the question: they are Lorentz transformations for a given observer in inertial motion.
Notice that E, B, H and D are related by those transformations while M and P no, at least in a simple form. This is a problem because M is a part of B and P of D.