Most direct route from Georgia to Kazakhstan. The part close to the Caspian Sea under Astrakhan has no road, instead there's a 300km stretch of sand and clay trails which can be technical in wet conditions. Expect loose sand in hot summer time.
There are plenty of checkpoints at the Russian/Georgian border area which can take up a bit of time. Regardless, if you have time constraints, this route is the fastest. Not advised for +180kg bikes.