From the monster compendium:

Regeneration: The troll regains 10 Hit Points at the start of its turn. If the troll takes acid or fire damage, this trait doesn't function at the start of the troll's next turn. The troll dies only if it starts its turn with 0 Hit Points and doesn't Regenerate.

The magma from the volcano is a legitimate threat to them, since it is hot enough to set organical matter in fire.

All you need to do is chopping them into little bits and loading the parts onto one of these:

They won't die of starvation, so time is not an issue. Just take your time to whatever destination you have in mind. You will need to keep grinding the little pieces on the way otherwise they will regenerate into full grown trolls.

Once in the destination just set them free and watch them regrow. You may even have more trolls now since you helped them reproduce:

Fission, in biology, is the division of a single entity into two or more parts and the regeneration of those parts into separate entities resembling the original.

First of all, harvest a good amount of sedative. However you transport them, the least thing you want is having 100-200 tons shaking your carrier.

Then you have two options:

1. 
   Antonov An-225 Mriya: with a Max takeoff weight of 640,000 kg and a cargo hold 43.35 m (142.2 ft) long × 6.4 m (21 ft) wide × 4.4 m (14 ft) tall, you can use it to transport one troll at a time, traveling at 800 km/h to a range of 4000 km.


2. 
   TI class supercontainers, currently the four largest ships in the world: they can each carry 3166353 barrels of oil, which roughly correspond to 392627772 kg, or 432798 short tons (about 4000 trolls), at 30 km/h.

I hope it is clear that option 2 is the most effective: the Antonov would require 10000 trips, while the TI class could do the job in just 3 trips (there are 4 of them, how lucky!).

"Just" make sure that the trolls are sleeping for the entire length of the trip. Those supercontainers are designed assuming that the oil won't smash the walls in hanger.

How you get to put them on the carrier is another problem, and for sure I cannot ensure 0 casualties.

Container ship, Ro-Ro, bulk carrier... whatever ship(s) you can get cheaply and quickly...

For long term transport instead of using elephant tranquilliser (Carfentanil, is a risky opioid), use instead high dosage of first generation antipsychotic agents. They are cheap, (generic) tend to calm down otherwise furious patients very effectively and are mostly safe for long term use. They were tested on wild animals for facilitating their transport and seem fine for this purpose too.

(considering nasty side effects, better go away before those trolls get off those meds...)

This won't work.

At least not with the speed you want. But your best bet is...

Aircraft carriers

(ref)

The United States has 10 Nimitz-class aircraft carriers. Each is designed for 6000 people. If the average person is 65 kg, that is 390,000 kg just in people weight, or 430 short tonnes. Let's call that 3 trolls.

Each carries 90 aircraft in the 6,800 kg range each. 6,800 x 90 = 612000 kg = 675 short tonnes = 5 trolls.

Take away other things the aircraft carrier can do without for a bit and add back some people and you're looking at about 10 trolls per trip per vessel. Or 100 trolls per trip for the 10 vessel fleet. And those 10 appear to be it for full fledged aircraft carriers, though there are 9 other smaller carriers.

Other countries have smaller ones. Russia has 1. UK none right now. France 1. India 2. Japan 3. Australia 2 (or more?). Italy 2. South Korea 1. Brazil 1. Spain 1. Thailand 1. For a total of 15. Assume an average of 6 trolls per trip (just a guess) and that's 90 more trolls per trip.

This brings our per trip total to 190 trolls (an estimate of course). Assuming that every country with an aircraft carrier is willing to loan them out for this purpose (after offloading all the planes).

It would take 52 trips to transport all 10,000 trolls off the island. Not counting their gear. And assuming they're willing to get on to the carriers. Which you've made clear they are not.

1. Neutralize the volcano! This might be simpler than moving 10,000 massive, grumpy trolls.




2. How about a bunch of blimps/dirigibles? They can lift a couple hundred tons, and can get good top speeds in the right conditions, much faster than a ship low in the water with troll weight. Depending on how far it is, you might be able to drop off the sleepers before they wake up.




3. If they have a huge snorkel, can these guys survive walking on the ocean floor? They sound very tough and dense. It sounds like they don't like water much, but they might like water more than lava when the time comes.

Are your trolls stupid? Can they communicate with humans?

If the answers are no, and yes, then simply tell them. Make announcements through speakers on helicopters that the volcano is about to go and there are ships nearby for rescue.

If the answers are no and no, then simply have the ships make noise to draw their attention to the ships and wait for the volcano to erupt. In this case, keep the ships away from the shore till the volcano goes off. Remember to invest in light vessels to ferry the trolls to the ships. How to arrange the transfer from shore to lighter and lighter to ship is an exercise left to the rescuer.

If the answers are yes and no, then you're out of luck. It can't be done without a portal or something similar.

Don't move the trolls - move the volcano instead. Just blow some undersea nukes near their island until the magma can flow out there. Tsunamis are no problem, as according to Renans answer trolls can only die when in acid or on fire.

As far as I know, this has never been tried before, so make sure to bring a lot of nukes!

If you really want to move the trolls, and to do so as quickly as possible, ballistic trajectories seem to be the option of choice. Stun them, pack them into fireproof clothing with a fixed iron belt and shoot them out of a railgun.
According to this with a 45° launch angel and an initial velocity of 1000m/s you could launch your trolls around 102km away from their home island. To achieve this, you would need a lot of energy:
E=0.5*m*v²=0.5*150000kg*10^6m²/s² = 75GJ

Since you only have 48h until the trolls wake up, you'll need a power of
P=E/t=75GJ/(48*3600s)=434kW

with an efficiency of let's say η=0.2, we get
E_eff=E/η=375GJ and
P_eff=P/η=P*5=2.2MW.

Producing this energy via a gas turbine reactor, you have a conversion efficieny coefficient of η_th=0.33. For the resulting thermal energy we get
E_th=E_eff/η_th=1.125TJ
Using the probalby easily available crude oil as fuel (which has E_s=45MJ/kg), you'll need
m=E/E_s=25000kg=25 metric tonsof oil.

This would only cost around 14000 USD, though the reactor, the energy storage, the railgun itself and 100000 fireproof flight suits would probably be much more expensive.

If you like this idea, I could calculate some more details...

Everyone's coming at this the wrong way. You're trying to move them a long distance quickly. The fastest way to handle this is ignore the implied requirements of the question--don't move them halfway around the world, move them to the closest uninhabited island.

Your vessel of choice is the Russian Mi-26 helicopter. It can lift a troll with capacity to spare, a bit over 300 have been built, I don't know how many are still in service. Trank a troll, lift it to the nearby island. 40 flights should do it, you should have them off in a week or two.

Now you still have to transport them to their final home but you don't have the volcano breathing down your neck while you do it.

Once they have had time to recover from the initial dose you trank one again and lift it to a ship. There it's fitted with some very strong and immobilizing restraints. Plenty of ships are capable of hauling 30 ton objects around, getting them to their new home isn't going to be a problem. The restraints are fitted with explosive bolts, you lift them onto the island and fire the bolts. (Note: Explosive bolts do not actually go boom, the charge just rips the bolt from it's anchorage. Whatever trivial damage it suffers when the bolts fire will quickly regenerate.)

This question has been answered here:

https://movies.stackexchange.com/questions/63545/are-helicopters-capable-of-carrying-this-type-of-giants

In the movie The BFG giants are carried by helicopter. The answer on movies stack concludes existing transport helicopters could do the job. I think the trolls in the question may be slightly larger than the movie giants are calculated to be, but 2 helicopters can team up to lift cargo suspended beneath in a net, as the giants are. I found images of helicopters carrying a jumbo jet in this manner.

As re sedation I suggest ketamine.

http://www.cesar.umd.edu/cesar/drugs/ketamine.asp

Ketamine is considered a dissociative anesthetic. This means that the drug distorts the
users perception of sight and sound and produces feelings of
detachment from the environment and ones self. The drug also has
anesthetic properties that have been used in both human and veterinary
medicine, and is currently a Schedule III controlled substance. For
humans it has been used in radiation and burn therapy, treatment of
battlefield injuries, and for children who have adverse reactions to
other anesthetics.

Ketamine is safe as regards cardiopulmonary toxicity and it is also an excellent veterinary drug if the trolls are closer to animals. They will be awake, breathing, but just chilling out in the net. They will be disinclined to struggle and hurt themselves.

Also, 10,000 is a lot of trolls. Triage will be necessary. To save the species, you will want to prioritize juveniles - easier to move, possibly less aggressive and with a longer reproductive life. Adult rescue should prioritize mothers of dependent juveniles.

Large trolls could be moved via drone boat powered floats. Here is a float made of 4 blocks of polystyrene foam adequate to move a 31,000 lb dam part.

https://univfoam.com/floatation

You would need more blocks to float a troll but that is fine. Floats can be cheaply assembled with off the shelf parts and attached to drone boats. A satellite uplink will allow random available boats to be converted to drone boats. They will moor off shore. Mockup dummy trolls can be positioned aboard some of the floats so trolls on land will see that trolls can ride these things. When the volcano erupts, trolls will move towards safety. They cannot swim but they are tall and they can wade and the floats are within wading distance. Even if they have not waded, the dummy trolls will hopefully inspire them. They will wade out to the floats and climb on. Once there is a troll passenger the drone boat will head out to sea with troll in tow.

At sea, trolls who jump around will fall off and drown. Each vessel has only 1 troll passenger at risk. Vessel / float failure means that passenger drowns. Calm and lucky trolls will make the ride in peace.

If the trolls are too stupid to figure that out, then those remaining after the airlift will not be rescued. If they wade out and destroy the float, too bad for them. If the volcano erupts with such violence there is no time to seek safety, the leftover trolls die and you might lose your floats and drone boats. No humans will be at risk. Cheap boats and cheap floats can be used: low risk, high yield.