Nothing more than connect an Ethernet shield 2 to a Due and use the appropriate libraries (obviously). Nothing more, nothing less, no guessing required. This was a simple compatibility question.
I did try googling for the answer, didn't find it and figured the forum would be a good place to ask.
If I had asked (on an AV forum) if ABC blu-ray player was compatible with DEF TV, I wouldn't expect to provide information on what disks I might be using and how many people are watching.
After mentioning electrical compatibility in my previous reply, I note that this is still unanswered...
Given that the Due is 3v3 and standard Arduinos are mostly 5v, for which most shields are primarily designed, this is probably the most important compatibility issue.