It is clear that the Don used tachankas quite a lot. We have witten accounts of them, and the cavalry units all have MGs in their orders of battle.
But clear pictures of them, which we can definitively place as Don, are very rare. Fortunately, the British Imperial War Museum has a couple taken by their military mission to South Russia.
The first clearly shows that some MGs were carried on two-wheeled military carriages. This is not my area of expertise, but these look like British ammunition wagons. The British shipped a lot out to Russian, and they are practical and sturdy. (They are also possible for gamers to find.)
The next photo is a more standard Russian cart, three horse I think.