The immigrants rescued from a shipping container yesterday are Sikhs who are understood to be from Afghanistan, it has emerged.
The 35-strong group's arrival at the Port of Tilbury yesterday prompted an international investigation after one man died.
Superintendent Trevor Roe, of Essex Police, said: "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.
"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."
A post-mortem examination into the man's death is being conducted today while the container in question is being forensically examined.
Police believe substances found in the container are cleaning chemicals which are not thought to pose any health risk.
Thirty of those found screaming and banging in the container have now been released from hospital, with the remaining four likely to be discharged today.
All will be taken care of by the Border Force in a "suitable location" once officials have finished questioning them. The Red Cross provided food and welfare for the group overnight.
The port has resumed business as usual, while police said concerns that a further container with people inside may have arrived at Purfleet were later dismissed.
The company which owns the container has not been named.
Belgian police earlier indicated that they believe the immigrants were probably already inside when it was dropped at a European port before setting sail for Britain.
Peter De Waele, spokesman for the Federal Police in Belgium, said it appeared to be "impossible" for the 35 men, women and children to be loaded into the unit in the time it was at Zeebrugge, a port in the north of the country.
He said investigators have been combing CCTV and are "very hopeful" that they can track down the driver and company of the vehicle that deposited the container.
De Waele told the Press Association each container is captured on camera and their seals are subject to "very strict" controls.
He said: "We saw that this container was there for one hour in all in an area where there is a lot of cameras, a lot of people and a lot of cars driving. It was around 6pm so it wasn't dark.
"We think that the possibility [is that] the victims were already on the container before it was put down in Zeebrugge because in that area it is rather impossible to put 35 people in that container.
"That is not exact information. I think the investigation has to give answers to this question."
He declined to confirm whether detectives have already identified the vehicle which delivered the container to Zeebrugge.
But he added: "My colleagues told me that they were very hopeful looking at the pictures that they [could] find the company and also the driver who put the container in Zeebrugge.
"It is too early saying the driver is involved but when we find the driver we can work backwards."
De Waele added that they are working closely with Essex police and described tackling human trafficking as a "priority" in Belgium.
The container was one of around 50 on board the P&O Ferries commercial vessel the Norstream when it arrived in Essex.
The immigrants were rescued after port authorities heard noises from inside at around 6.30am yesterday.
Bhai Amrik Singh, chair of the Sikh Federation (UK), said: "It is a disgrace the persecution of the tiny minority of Sikhs from Afghanistan has largely been ignored and it takes an incident like this to remind us all that they are also being exploited by human traffickers."