The relative in question (marriage rather than blood), is said to have fought with Pershing in the Villa Expedition into Mexico beginning March 15, 1916. The Mexican flag is there, opposite the US flag, and the centerpiece of the work is an eagle sitting on a cactus while devouring a serpent, part of the fine details of the Mexican flag, based on the Aztec symbol for Tenochtitlan (now Mexico City), recalling an old Aztec legend about the founding of the fabled city.
The strange part is: the citation says it is in "...Appreciation of the service rendered by you in France in upholding the highest traditions of the Marine Corps, and to congratulate you upon receiving this honor conferred upon you by General Pershing."
Grandad signed up when he was underage and fought anyway.
It seems possible he could have fought under Pershing in Mexico in 1916, then followed him to the trenches of WWI. He fought in the Battle of Belleau Wood, and came home a changed man.
Anyway, the Citation is made out of actual feathers. If it's real, it belongs in a museum. I have never seen anything like it.
What you have is certainly an interesting piece (museum curator/collections manager here). I've seen art of the period with feathers similar to that before, but certainly nothing dealing with the military, so I'm finding this interesting.
We could use a little more information to go on.
If none of that ends up helping, the next best thing to do would be to contact the National Archives and see if you could obtain your grandfather's military service records. It's easiest for the direct next of kin to do it (son, daughter, surviving spouse, etc), but you can always contact them and see what (if anything), they can do for you. Those records would (in theory) show whether he served in Mexico with General Pershing, and any awards or medals he received (I will say that this certainly doesn't appear to be an official military award.)
Don't be afraid to contact the archivists at the National Archives, if they can help you they'll certainly try. Don't think of them as bureaucrats, think of them as history nerds who have to follow bureaucratic rules. Please note there may or may not be a nominal fee for copies (probably under $100).
Let me know if you post any more pictures, or have any more information you are willing to share. If you end up finding more about the piece in the future, feel free to come to /r/MuseumPros, and we'll see if we can find an appropriate museum or archive for it so that the piece can be preserved and available for study.