Russia was recognized as the automatic successor of the USSR at the UN etc. Ukraine and Belarus also had enjoyed special status since the Yalta compromise, so they were members of the UN since 1945, thus had their names separately recorded on a bunch of treaties, already.
But that was not the case for the other Soviet republics. I'm picking Uzbekistan here as hopefully representative of the remainder (as it has the largest population among these, surprisingly a bit larger than that of Kazakhstan, which has far wider area though). So how did a country like Uzbekistan handle international law in the aftermath of declaring their independence? Clearly they had to apply to join the UN and similar organizations. Did they also have to separately sign and ratify themselves every single [multilateral] treaty of import that they want to continue applying that the Soviet Union had been a part of, from those regarding road traffic to those related to prisoners of war?
N.B., on quick google search, it seems in some respects there was a degree of automaticity:
In its decision on jurisdiction, the arbitral tribunal in World Wide Minerals v. Republic of Kazakhstan held that Kazakhstan succeeded to the obligations of the Soviet Union with regards to its 1989 Agreement with Canada on the Promotion and Reciprocal Protection of Investments.
But how far did this automaticity go?