The boys, who are aged between 11 and 16 years old, along with their 25-year-old coach Ekkapol Chantawong, went to explore the Tham Luang Nang Non cave after football practice, but did not return home that night.
Somboon Sompiangjai, 38, the father of one of the trapped boys, said parents were told by rescuers ahead of Sunday's operation the "strongest children" would be brought out first.
When it was realised that the boys were in the cave system, divers began the process of trying to find them in the enormous area.
Prime Minister Prayuth Chan-ocha went to the cave to inspect the rescue operation later on Monday, with navy rescuers giving him a rousing cheer.
"The operation is being supervised by the Thai authorities".
He said it's expected to take many hours - and could take days - for all 12 boys and their coach to emerge. Medics appeared to remove one person on a stretcher but hid the person's identity behind multiple white umbrellas.
A team of 19 Australian divers are assisting with the rescue operation.
Onlookers cheered as ambulances transported the rescued boys from the Tham Luang cave complex on Sunday.
Around that time, an ambulance left the cave area and headed toward a helicopter, which landed at a nearby hospital.
The children's families have not been told which boys have been rescued.
The boys and soccercoach trapped in a Thai cave have written a letter saying they're OK.
Among these were that none of the boys had scuba diving experience, and that they could easily panic while swimming underwater in darkness.
The second day of the underground rescue attempt began at 11:00am local time (2:00pm AEST).
The head of rescue mission Narongsak Osottanakorn speaking at the news conference.
Eight boys are still inside the cave and along with the team coach.
Rescuers have been unable to extend a hose pumping oxygen all the way to where the boys are, but have brought them some oxygen tanks.
Experienced cave rescue experts consider an underwater escape a last resort, especially with people untrained in diving, as the boys are. He ran out of air trying to swim out of the cave.
He said that expert divers who had taken part in Sunday's operation needed to rest at least 20 hours, which would have delayed the start of the second phase to between 8pm and 10pm on Monday. Osatanakorn said hospital officials are working on a plan to allow parents to see the boys "at a distance through glass". It takes several hours.
"Any way we can help involve them in their own rescue is beneficial", he said.
"The water level is not worrisome ... yesterday's rain did not affect water levels inside the cave".
1 NEWS' Kimberlee Downs has the latest from Chiang Rai.
Narongsak reportedly added that rescuers' plans had been created to save only four people at a time, suggesting that one member of the soccer team, or the coach, may have to stay behind after the third phase of rescue operations.
Elon Musk's Space X rocket company is testing a "kid-sized submarine" that could be sent to help boys trapped in a flooded Thailand cave.
Musk posted videos on Twitter of the aluminum sub being tested at a swimming pool Sunday midafternoon California time.
He tweeted Sunday evening the group was en route to Thailand with the device and would reach the destination in about 17 hours. "Their plan is designed for rescuing four".
Witnesses say the boys freed Monday were treated at a make-shift hospital at the site, before being taken to a local hospital.
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