I'm not sure about MG42/DP28 on-the-move stats, if they are "elite" or anything like that, and how they compare to Commando Bred and Para 1919.
Slightly off-topic, but that's another weird thing - there are some inconsistencies between normal and fire-on-the-move versions of LMGs. First of all, there are actually 3 weapon profiles for LMGs: regular (one that can't fire on the move), moving (one that can fire on the move) and moving_no_prone (only Obersoldaten get this one and as the name suggests - it can fire on the move and has disabled prone-firing).
For M1919A6, their accuracy compares as follows: regular > moving > moving_no_prone. There's a sharp drop in accuracy between the last two and the regular version fires slightly shorter bursts.
For LMG-34 it's: moving_no_prone = moving > regular. Additionally, regular deals only 4 damage compared to 6 of the "elite" versions.
For LMG-42 it's: regular > moving = moving_no_prone. Both of the moving versions have slightly shorter reload on average.
For DP-28 it's: regular > moving = moving_no_prone. This is the only one without any hidden "bonus features".
Vickers K and Bren obviously can't be compared in this manner.