This is from Edge of Empires by Donald Rayfield who is a top expert on both Stalin and Georgia.

In the 1920s, Stalin rarely visited Tbilisi, and there was no shortage of tortures and brutality in the region, much at the hands of Beria, who was also Georgian. At the time, Georgia was forced into the Transcaucasian Federative Republic with neighboring regions. In the 1920s and 30s, Stalin purged Georgian subordinates in Moscow, including his brother in law, Aliosha Svanidze.

However, Beria spared a few Georgian writers because Stalin had read them. Then in 1938-1941, Stalin took a renewed interest in Georgian history, culture and language. At the same time, Georgia also enjoyed a "rationalization of tyranny." Stalin appointed Kandid Charkviani, a humane guy, to lead the Georgian party in a move which was designed to limit the actions of Beria. Charkviani removed torture chambers, reduced executions, and lessened censorship. "In 1939 Tbilisi seemed strangely liberal." Also there were many Georgians placed in positions of power in Moscow and elsewhere. In the later period of WWII Stalin negotiated with de Gaulle to recover crates of Georgian treasure and artifacts which had been owned by the Mensheviks.

However, Stalin's attitude was ambiguous after the war; he was interested in the culture but didn't see his own ethnicity as Georgian. He referred to the people as "you Georgians". There was a lot of brutality and crackdown in Georgia at Stalin's hands after the war. Charkviani was removed from his position but not killed.

Beria was actually more favorable to Georgia after the war than Stalin. Beria was seen as a "Georgian first and Soviet second" and distrusted. Stalin conducted the "Mingrelian Affair" which was a purge of local Georgian Communist leaders in a move against him. Stalin's death was mourned in Georgia but Beria's accession after him was cheered and during his short time in power he was favorable to them.

So as you can see there seem to be a lot of contradictions here but that's probably what you get in a decades-long period of rule in a state as schizophrenic as the USSR. I don't know anything about the rest of the USSR so I can't really say anything to compare it to what happened elsewhere.