Thank you for your comments and your POV, Throbert McGee!
It definitely adds to the discussion.

"заявило, что не будет препятствовать своим добровольцам выехать..." -- it seems highly unusual for a government to say, "we won't prevent our volunteers from going out to join the fight", as though they have no control over where the military goes and what it does!
"Добровольцы" are not necessarily military men, it could be any civilians who volunteered to help of their free will. But it was manipulation on a Soviet part, of course, since no one could leave the country without state approval, civilian or not, and a statement that no one would be "discouraged" from helping was almost the same as saying outright that these attitude would be silently encouraged.

"В ночь на..." is neutral, it means that something happened between 0 am and roughly 5-6 am. So "under cover of darkness" is probably too poetic a translation.