In the game units gets little white dots beneath their icons, I think it mark something like fatigue or bad supply but I can't find proper info about them... what exactly do they mean?

In my rather, so far, limited experience, if the white dot is in the lower left-hand corner, that denotes the unit is "sleeping", and to be skipped in succeeding turns, till it is awakened. Both these things are done by the human player. To make sure, right-click on the unit, and a menu should pop up, indicating its status, which you can designate.

Hi borsook

These sound like their experience levels, could that be what you're seeing here?