The second day of the underground rescue attempt began at 11:00am local time (2:00pm AEST).
An Australian doctor with cave diving experience examined the boys on Sunday and declared them fit for the operation, authorities said.
Interior Minister Gen Anupong Paochinda said they are generally "strong and safe" and under close medical supervision. They will be shuttled to helicopters that will take them to a hospital. They say rescuing the eight remaining boys and their soccer coach could take up to four days.
"In a press conference, head of the rescue Narongsak Osottanakorn said the operation got underway sooner than expected as "[Sunday] was the best situation - in terms of kids' health, water and our rescue readiness".
The video also showed a Caucasian diver walking heavily up a muddy slope carrying a face mask and a small oxygen tank, wearing a backpack with his flippers dangling behind his legs.
The group took less than four hours to arrive at 2pm (8am United Kingdom time) in the cavern where the boys have spent 15 days nearly three miles from the entrance.
"I would like to inform the public at home and those who have been giving us support all along, after 16 days, today's the day we've been waiting for, we are seeing the Wild Boars in the flesh now". This week, a former Thai navy diver died during one of the rescue missions. He said that process can take several hours.
Known for his ambition and sometimes insane ideas, Musk promised to develop a submarine to aid in rescuing the boys.
Thailand's Meteorological Department said there was a 60 percent chance of rain Monday with thunderstorms forecast throughout the week.
The entire nation is glued to the media coverage of the rescue mission, and Thai authorities have insisted they will not compromise on the safety of the trapped group. After the four boys were removed from the cave, heavy rain started falling.
The boys and their coach, whose team is known as the Wild Boars, became stranded when they were exploring the cave after a practice game on June 23. Rising water levels, too, as the result of heavy rains is also an issue.
The ordeal has riveted Thailand and captured the world's attention.
America and Britain joined the local rescue team, and sent in more than 30 American military personnel alongside pararescue and survival specialists.
Elon Musk posted a video of people testing a mini-submarine in a swimming pool to aid the rescue of a youth soccer team in Thailand.
In letters written by the boys released Saturday, the boys appeared calm and reassured their families they were "fine".
Handwritten notes by the children trapped in a cave read, from the top: 'Please don't worry Dad and Mom, Biw has been gone for two weeks, I will help mom sell her goods every free day, I'll quickly make my way.' The message below reads: 'Please don't worry, be happy.
"I promise to take the very best care of the kids", he wrote.