Firstly, how do they start moving with no clutch? Push the bike a rolling start and drop in gear?
You'll likely have to take the clutch apart and see how the basket and plates are. Here is how the clutch works:
You pull the handlebar lever which pulls the cable. When the cable is pulled, it turns the mechanism on the engine which pushes a rod that goes through the engine into a ball bearing. This ball pushes on another rod which disengages the pressure plate from the clutch pack (series of friction discs and metal plates). It is this clutch pack (which sits in the clutch basket) that transfers the power from the engine to the transmission and drive sprocket.
When the clutch is engaged, the pressure plate is pushing against the clutch pack due to the clutch springs. When the cable is pulled in, the pressure plate is pushed away (through the series of mechanisms and rods) and therefore disengaged so the engine is able to spin while the drive sprocket does not.
That's the simple gist of it. So if the mechanism turns on the engine, it's possibly any of the things listed after that. The rod could be broken, the ball flattened and perhaps welded to the rod or even missing, the clutch pack could be worn or could be as simple as the discs and plates that make up the clutch pack are 'stuck' together. When they are stuck together, the pressure plate can be disengaged but the discs are physically stuck to the metal plates that turn with the engine. This continues to turn the drive sprocket even though everything else is working fine.