Everybody who will look at your diorama will see the Gothas, the fantastic truck, the figures (any progresses here in sculpting?) and never ever would ask, if the horse drawn wagon fits the scene - it is just such a small detail... Anyway I spend quite a while in researching German horse drawn equipment of WWI which is annoyingly bare of good sources, books are very rare and this issue seems not to attract many people, which I completely understand. What is a cart against a beautiful airplane !
I don't know why, somehow I like these horse drawn wagons anyway especially their fallen-out-of-time appearance next to a more modern fighting machine (like tanks or so). And there's really cool stuff like aircraft repair wagons, or water desinfection wagons, look like some Steampunk stuff...
So thank you very much for posting these pics, they are really valuable! But in fact they all seem to be standard German Army equipment, even the civilian style wagons. I can see many leichte Feldwagen, some heavy Feldwagen (e. g. First pic, on the extreme Right side), and the wagon on the first pic with the hood is an official doctor's wagon. For some of the carts I'd have a look at my books - but back to the original question: I totally agree that they used civilian wagons at the end of the war, but I don't think that they took the extra work to paint them field grey and add a coat of arms. But of course it's not impossible...!

Normally they used to take heavy field wagons for transporting ammunition, luckily Italieri offers a good kit:

The "Hf.2" (these designations were introduced after ww1) is in fact a Schwerer Heeresfeldwagen, as you could use mobile kitchens of ww2 for ww1 "Gulaschkanonen". But sadly I have no photos at hand, that shows the former ones at Bomber airfield. I'll check my references at home asap.
Best wishes
Borsos