A trans-axle gearbox has two shafts inside of it, the one that comes from the engine and another that is driven by the gears going to the wheels. The gears are engaged by a selector pushing the dog gear into the driving gear. Before the gear is engaged the synchromesh slows the driven shaft down to the same speed as the gear so that the gears can engage with out clashing. Once a gear has been engaged the wheels turn at the speed that its determined by the gear ratio of the driving gear and the driven gear.
It is important to use the right lubrication for a gearbox. If the wrong lubrication is used like differential oil it can cause the synchros to slip and that would result in the gears crunching it could also produce more wear on the gears.
Good intro to the Gearbox blog
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