My opinion is that there are probably only a few manufacturers of all of the brands out there. I would guess that WalMart uses one of the major oil companies to supply the grease with the WalMart label on it.
My best guess would be that as long as the specifications are the same there would be no problem. I think that the main thing is the frequency of the greasing. By putting in new grease you drive out the metal dirt and moisture that has accumulated in the bearing areas.
I personally grease untill I see the grease come out without any discoloration, then I opperate the FEL and give each zerk another couple of shots. It is also a good idea to wipe the zerks off before greasing so that no dirt gets pushed into the area that you are greasing.
Just my opinion though,
Don