They were discovered yesterday after a freighter arrived at Tilbury Docks in Essex from Belgium.
One man was found dead and the others were taken to hospital to be treated for severe dehydration and hypothermia.
The 35 people found inside the container are understood to be Sikhs from Afghanistan, Essex Police said.
Earlier, the police described them as people believed to be from the Indian sub-continent and questioned some of them.
A homicide investigation is continuing into the death of the one person.
After they were treated for severe dehydration and hypothermia, they were taken to a detention centre.
Thirty have now been released into the care of police and Border Force staff. Four remain at Southend Hospital. It is thought they will be discharged later.
A post-mortem examination will be carried out on the body of the deceased and the container forensically examined.
They are being held under UK immigration laws and it is expected they will be interviewed through interpreters at the immigration centre near Tilbury.
Essex Police described the men, women and children as victims of "people trafficking" and are working with Interpol and other international authorities to try to establish what happened.
Many of the survivors spent the night in hospital but none is thought to have suffered serious injury.
One man is believed to have been questioned by officers already.
India's Ministry of External Affairs has confirmed being in contact with UK authorities to ascertain their nationality.
The discovery was made after the container arrived from the Belgian village of Zeebrugge early on Saturday morning when "screaming and banging" were heard coming from inside.
"It is a homicide investigation...we will be looking to see where the origin and the gangs or whoever may (be) involved in this conspiracy to bring these people in this way over to this country. Clearly we need to try and bring them to justice," Supt Trevor Roe of Essex Police told reporters.
"The welfare and health of the people is our priority at this stage. Now they are well enough, our officers and colleagues from the Border Force will be speaking to them via interpreters so we can piece together what happened and how they came to be in the container," Roe said.
"We now understand that they are from Afghanistan and are of the Sikh faith. We have had a good deal of help from partners within the local Sikh community in the Tilbury area to ensure that these poor people, who would have been through a horrific ordeal, are supported in terms of their religious and clothing needs," he said.
The Red Cross provided food and welfare for the group overnight.
Belgian police said they believed the lorry which delivered the container in Zeebrugge had been identified through CCTV footage.
Chief Inspector Peter De Waele, from the Belgian police, said: "The Belgian civil police is checking all the images and my colleagues are very, very hopeful that we find the truck who put the container in Zeebrugge.
"I think it is very, very important that we have the identification of that truck and also of the driver."
It is not known where the container, one of 64 aboard the P&O commercial vessel Norstream, originated, nor where the people inside it were heading.
"It shows how desperate people are to improve their economic situation how desperate they are to leave their own homes, and own countries, and hope to arrive in somewhere that's more accommodating, more kind, and offering them a better quality of life. Usually, they're sadly wrong," said Anthony Steen, chairman of the UK's Human Trafficking Foundation.
Media reports around the time of Afghanistan's elections earlier this year suggested the total number of Sikhs in the country was about 3,000.
There had been about 200,000 in the 1970s but civil war in the 1990s and the rule of the Taliban caused most to flee.