Ah, sorry I get confused with the multitronic, s-tropic, Tipttonic jargon. The multi-plate wet clutches are not used as you would use a clutch in a manual transmission, they are only used to engage a planetary gearbox to enable reverse. So in short no, these will not be the problem.

The way the CVT works is by having two sets of opposing cones and a special chain wedged in the middle which is know on as the variator. The chain only transmits torque if it is compressed (similar to a clutch). The pressure in between the cones is controlled by a hydraulic valve, giving you your constantly varying ratio. However, when at a standstill, the chain is allowed to slip on the cones so that it acts like a torque converter, then when you pull away the cones are clamped together with more pressure and the chain locks.

When you put it into tiptronic mode, what happens is it will work in defined steps of ratio and apply more pressure to keep it at that ratio and only change when you hit gear up/down. So this is why you don't see slip when in tiptronic. I believe in sport there is just more pressure on the cones to transmit the increased torque, again why you don't see slip in sport mode.

Now I have explained how it works, your problem could be a few things:
1) The chain that links the variator set could be worn and it not locking properly, so it is slipping and not transmitting torque and your engine rpms are flaring. (Solution: Transmission Rebuild)
2) Old oil is causing poor hydraulic performance. Meaning the transmission is sending what it thinks is enough pressure to lock the variator, but due to the gloopy oil it is not enough and consequently the variator chain is slipping and your engine rpm is flaring (Solution: Transmission Oil Change)