The group’s arrival at the Port of Tilbury on Saturday prompted an international investigation after one man died.
Police in Belgium were last night reviewing CCTV footage of the shipping yard from which the container had been shipped to the UK from to establish who had dropped off the cargo.
The group of men, women and children were heard screaming and banging from inside the container by port authorities in England and having primarily been concerned for their welfare, Essex Police said they had now begun to piece together their ordeal.
Superintendent Trevor Roe said: “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 was being conducted yesterday and the container was being forensically examined.
Police believe substances found inside the container are cleaning chemicals which are not thought to pose any health risk.
Thirty of those found in the container had been released from hospital yesterday afternoon.
All 34 people will be taken into the care of the Border Force once officials have finished questioning them.
The Red Cross provided food and welfare for the group overnight after they were discovered at around 6.30am on Saturday.
The company which owns the container has not been named.
Belgian police earlier indicated that they believed 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 was unlikely the 35 people had been loaded into the unit while it was at Zeebrugge, a port in the north of the country, as each container is captured on camera and their seals are subject to “very strict” controls.
“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,” Mr De Waele said.
“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.”
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.”
The shipping container was one of around 50 on board the P&O Ferries commercial vessel the Norstream when it arrived in the Essex port on Saturday morning.