Wikipedia has a good animation of a manual transmission, but something irks me.
If I understood correctly, in the animation, the input shaft is in green, and the output shaft is in cyan. The dog clutches (synchromesh in modern cars) are therefore between the free gears and the output shaft. This forces both the engine and the gear assembly to change speed in a gear shift. If the synchromesh were between the input shaft and the free gears, only the input shaft (therefore the engine) would need to change speed during a gear change. I don't see anything after a quick Google that suggests things are different in many other gearboxes.
What is the reason for this that I'm apparently missing? Acting as a flywheel is the only that crosses my mind.