Presumably they mean propagating EM radiation we observe from earth appears to behave the same on earth as we observe from distant galaxies since the event that created them happened at a time much different than ours and a distant region of space.

I mean, technically the EM radiation we observe from distant galaxies does look different than the EM radiation we observe locally: it's red-shifted.

I'm sure someone has proposed this is due to physical constants changing over time, rather than the expansion of space-time, and I'm sure someone else has explained why this is wrong.

The wavelengths of physical processes are equal. If fundamental constants changed, we'd expect, say, the Lyman series to change too.

Yes, we've measured comparable wavelengths. It's one way we can measure the red shift. Not just (red shifted) absolute wavelengths, but the relative spacing between them are quite sensitive to physical constants. These spectra can also be used for identifying the elemental composition of stars.

> What's meant by "the wavelengths are equal"?

Absorption lines of the elements in the stars whose starlight we observe. THey are the same after correction for redshift.