Police in the UK have said that the people found in a shipping container at Tilbury Port in Essex yesterday were from Afghanistan.
One man died and 34 men, women and children, aged between one and 72, were taken to hospital after they were rescued.
Officers said they were treating the incident as a "homicide investigation" after the man's death.
They are investigating how the group was trafficked into Britain and have been liaising with Interpol and other international authorities.
The unit arrived in the UK from Zeebrugge in Belgium early yesterday.
Many of those rescued spent the night in hospital after being admitted for hypothermia and dehydration.
The group was rescued after port authorities heard banging and screaming from inside the container at around 6.30am.
Describing them as victims of "people trafficking", Superintendent Trevor Roe of Essex Police said they had been in the container a "significant amount of time".