I am just wondering the 4 Maxwell equations (i.e Fadaray Law, Maxwell-Ampere) are Lagrangian or Eulerian description? Does it really matter?
The distinction between the two descriptions you consider assumes that there is a field of velocities describing the motion of the particles of the continuous body.
Adopting one or the other description respectively means (a) to use the initial position of the particles to label the integral curves of the field (Lagrangian description) or (b) to refer to the istantaneous positions of particles in a given rest space (Eulerian description).
The electromagnetic field does not provide this field of velocities so that the choice does not make much sense. Or maybe, to some extent we can say that only the Eulerian description can be adopted.
The EM field is not a continuous body or a fluid, even if it shares some features with these physical systems.