What are the mechanism and effects of non-linearity in an attenuating barrier in the presence of a high-strength EM field?
Non-linear material properties are wide ranging and depends on the purpose.

For magnetic fields, ferrite, cobalt etc are all linear until they start to saturate then the "mu" coefficent (permeability) can drop to a low slope value. This would look like this.


This one is steep when linear and flat equivalent to air (mu=1) when saturated.



If mu is high = 10k and drops to 1 then it is pretty significant reduction in inductance and shielding effects become transparent..  But in reality this would require a very high H field like a nuclear blast.  So then you need lower mu materials that dont saturate so easily like  thick wet concrete walls.

It all depends on the application.

There are lots of other non-linear properties such as rise time and aperture effect of a waveguide with 1/4 wavelength impedance inversion  effects.