Variated vespa models include the Bravo, Si, and Grande. The Ciao was also available with a variated transmission in one of the deluxe versions, though most were single-speed.
Kinetic also made a variated model called the Magnum, based on the TFR.
There are three primary components that are different between the two drive lines; the two pulleys that transfer power, via belt, from the engine to the transmission inside the hub of the rear wheel, and the transmission itself.
On a single speed bike the clutch is on the engine crankshaft pulley on the front, and there is a simple pulley on the input shaft of the transmission inside the rear hub.
On a variated bike the weighed variating pulley, or variator, is attached to the engine crank shaft. The clutch is on the rear pulley, which is a counter-sprung variating pulley. The transmission has a longer input shaft to accommodate this 'counter-clutch', which mounts much differently than the single-speed driven pulley. It is also geared differently. The single-speed and variated case bearing and gear layout is different; so a variated gearset cannot be installed in a single-speed case to create a variated transmission.
To do the conversion with parts from the same manufacturer, like variating a Vespa Ciao with Bravo/Grande/Si parts or a Kineitc TFR with Magnum parts, one can simply swap the clutch for the variator, and the transmission, driven pulley, and belt, for the variated ones.
Like most drive line parts they are inter-changeable between the two brands, and due to the relative scarcity of variated and performance Kinetic parts, Vespa parts are often used to variate TFRs (though there has been at least one Magnum variated Ciao), but there are a few caveats:
Vespa transmissions do not readily fit in Kinetic rear wheels; The splines on the shaft and the wheel hub appear to match, but are too tight a fit to slide in all the way. Usually people just use a Vespa wheel to match the trans, it is also possible to file the splines in the kinetic wheel hub down some. Using a kinetic wheel allows for use of the kinetic rear wheel bracket with it's sealed bearing and dirt lip seal to the wheel hub.
The newer Kinetic TFR-USA model engine crank/output shaft is cut differently than Vespa and older Kinetic TFR models, so Vespa and older Kinetic variators and clutches do not fit. This can be remedied by swapping to a Vespa performance, or even an older model crankshaft.
When purchasing a used transmission (either type), check to see if the metal seals behind the frame bolt holes have been compromised. This can usually be seen through the bolt holes, without disassembling the transmission. If the frame bolts are tightened without the split washer or without the transmission being mounted in the frame, the bolts will go in too deep and drive out the centers of the sealing caps (called welsh plugs or freeze plugs). If this happens to the lower bolt hole, the oil will drain out if the lower bolt is removed, or it may seep over time because the bolt design is not configured to seal fluids. Dirt and sand can also be introduced to the inside of the case if the bolts, frame, and case aren't clean during assembly. If the center of the cap is punched all the way out, the resulting slug may jam the gears and damage the transmission. The caps can be replaced if the correct size and thickness can be found, though this is no simple task.
The decompression valve may be required to start the bike after the variated transmission is installed. Single-speed bikes typically have enough belt tension to turn over the engine without using the decompression valve; variated and especially kitted bikes may not, allowing the belt to slip at the variator (the Malossi unit combined with a standard 'A' 1/2" wide belt is notorious for this). Keep this in mind when converting a bike from single-speed to variated. The Malossi performance belt for variated transmissions is reported to help ease starting, as are the gates ax and metric belts, which are 13mm wide, toothed, and lower profile like the malossi belt. Using 13mm rollers that are molded slightly over 13mm in diameter (such as leovince) also helps (worn out malossi or small-molded rollers of some other brands allow the pulley cheeks to come too far apart, allowing standard belts to bottom out on the center and slip, this is where the lower profile and toothed belts also help).