Did they flee to Moscow and act as governments in exile? Did they resist to the end along with the local Red Army units? Did they just disband and try to escape into the countryside?
I don't have a full answer, especially with how different parts of the old Soviet Union really were on a ground level.
I just finished Where Iron Crosses Grow on the Crimean area from 1941-1944, and since it is fresh I figured I would let you know for that area what a newer history says on the topic.
Many of the people living under the Soviet rule in the Crimea were ready for a revolt. They had been purged and starved and discriminated against by the Soviet government since its inception. Before and after the war huge portions of the population would be deported "trail of tears" style and Tartars were assumed by race to be traitorous to the Soviet system.
When the Wehrmacht advanced they did so quickly and a lot of people were suddenly behind or on the front lines. Remember its not just when the enemy passes you that transportation is screwed up. Supplies from your side to the front as well as retreating soldiers and civilians meant many people were stuck and starving all around the front line.
The war policy of Germany under Hitler's Commisar Order was easily used to eliminate anyone with any civil control in the Crimea because they basically had to be "party people" to get the jobs in the first place. The populace helped find and kill these former officials in many cases. There were even local units of Tartars supporting Wehrmacht troops. The local Crimean population actually faced less execution and terror campaigns under the Germans on a whole compared to Soviet rule before and after.
If an administrator did make it to safety deeper into the Soviet Union they could be accused as cowards and deserters as well as collaborators and subject to prison and execution. Also many of the leadership had been purged earlier during the 1930s so the people still in positions of power were usually the most fanatic in their ideology.
In Crimea during German occupation they still had the Crimean Autonomous Soviet Republic Council and still made plans for Crimea's future, but had little effect on any actual people living there until they recaptured the area. The Council in charge during occupation was either evacuated or tried and executed for "aiding" the enemy.