The 39 people found dead in a refrigerated trailer in Essex were Chinese nationals.
Police are continuing to question lorry driver Mo Robinson, 25, on suspicion of murdering the eight women and 31 men.
Officers in Northern Ireland have raided three properties and the National Crime Agency said it was working to identify "organised crime groups who may have played a part".
The trailer arrived in Purfleet on the River Thames from Zeebrugge in Belgium.
Ambulance staff discovered the bodies in the container at Waterglade Industrial Park in Grays just after 01:30 BST on Wednesday.
The lorry and trailer left the port at Purfleet shortly after 01:05.
Police said the tractor unit - the front part of the lorry - came from Northern Ireland and picked up the trailer from Purfleet.
Councillor Paul Berry said the village of Laurelvale in County Armagh, where the Robinson family live, was in "complete shock".
He said he had been in contact with Mr Robinson's father, who had learned of his son's arrest on Wednesday through social media.
"The local community is hoping that he [Mo Robinson] has been caught up innocently in this matter but that's in the hands of Essex Police, and we will leave it in their professional hands to try to catch the perpetrators of this," he said.
The lorry was moved to a secure site at Tilbury Docks on Wednesday so the bodies could be "recovered while preserving the dignity of the victims".
Essex Police said formal identification of the 39 bodies "could be a lengthy process".
Essex Police initially suggested the lorry could be from Bulgaria, but later said officers believed it entered the UK from Belgium.
A spokesman for the Bulgarian foreign affairs ministry said the truck was registered in the country under the name of a company owned by an Irish citizen.
He said it was "highly unlikely" the deceased were Bulgarians.
The Belgian Federal Public Prosecutor's Office said it had opened a case which would focus on the organisers and others involved in the transport.
A spokesman said the container arrived in Zeebrugge at 14:29 on Tuesday and left the port later that afternoon before arriving in Purfleet in the early hours of Wednesday.
It was not clear when the victims were placed in the container or if this happened in Belgium, he said.
St Peter and St Paul's Church in Grays will be open for people to light candles and say prayers between 12:00 and 14:00.
A vigil is being held at 18:00 outside the Home Office to "call for urgent action to ensure safe passage" for people fleeing war and poverty.
Shaun Sawyer, the National Police Chiefs Council lead for modern slavery and human trafficking, said while forces had prevented thousands of deaths, "tragically, for 39 people that didn't work yesterday".
He told BBC Radio 4's Today programme even if there were routes perceived as easier to get through, organised criminals would still exploit people who could not access those.
"You can't turn the United Kingdom into a fortress," added Mr Sawyer, who is the Chief Constable of Devon and Cornwall Police.
Thurrock's Conservative MP Jackie Doyle-Price said there needed to be an international response.
"We have partnerships in place but those efforts need to be rebooted, this is an international criminal world where many gangs are making lots of money and until states act collectively to tackle that it is going to continue," she said.
Richard Burnett, chief executive of the Road Haulage Association, said temperatures in refrigerated trailers could be as low as -25C.
He described conditions for anyone inside as "absolutely horrendous".
Prime Minister Boris Johnson said it was an "unimaginable tragedy and truly heartbreaking".