The 39 people found dead in a refrigerated trailer in Essex were Chinese nationals.
Police have been granted an extra 24 hours to question lorry driver Mo Robinson, 25, on suspicion of murdering the eight women and 31 men.
Three properties in Northern Ireland have been raided and the National Crime Agency is working to establish if “organised crime groups” were involved.
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 – entered the country via Holyhead in Wales on Sunday, having travelled from Dublin.
Speaking after a magistrate granted Essex Police more time to question Mr Robinson on Thursday, Deputy Chief Constable Pippa Mills said her priority was “preserving the dignity of the 39 people who have died and ensuring that we get answers for their loved ones”.
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 has been moved to a secure site at Tilbury Docks and police are due to begin the process of moving the bodies to a mortuary at Broomfield Hospital in Chelmsford.
They will be taken by private ambulance so that post-mortem examinations can take place, with the force expecting all the bodies to have been moved by the weekend.
Global Trailer Rentals Ltd confirmed to RTE News that it owned the trailer and said it had leased it on 15 October.
The firm said it had given Essex Police the details of the person and company they had leased it to.
Essex Police said it was the largest murder investigation in the force’s history and the victims were all “believed to be Chinese nationals”.
It said formal identification of the 39 people, one of whom is a young adult woman, “could be a lengthy process”.
China’s ambassador to the UK Liu Xiaoming tweeted that the embassy had read the reports of the deaths “with heavy hearts” and was in close contact with British police.
Lucy Moreton, from the Immigration Services Union, said the sheer number of containers coming into the UK every day made it impossible to look inside them all.
“We don’t have the facility to check the vast majority of freight which arrives in the UK, whether it moves or not,” she said.
She said disconnected freight containers were less likely to be searched unless there was “intelligence to the contrary that suggests we need to do that”.
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.
The Belgian Federal Public Prosecutor’s Office said the container arrived in Zeebrugge at 14:29 on Tuesday and left the port later that afternoon.
It was not clear when the victims were placed in the container or if this happened in Belgium, a spokesman said.‘
A spokesman for C.RO Ports, which operates terminals at Purfleet and Zeebrugge, said they would “fully assist” the police investigation.
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.
Police officers and councillors have signed a book of condolences which was opened at Thurrock Council’s chambers.
A vigil was held at 18:00 outside the Home Office to “call for urgent action to ensure safe passage” for people fleeing war and poverty.
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.