The Manston centre in Kent became overwhelmed in November 2022 at the height of the small boats crisis.
Andy Burnham wants it to justify its decision to put guards in towns and cities near an asylum base.
Councillors said they had been informed of the plans but prohibited from speaking on the matter.
The Home Office is considering using RAF Linton-on-Ouse to house asylum seekers.
Use our interactive tracker to explore the latest migration statistics for the UK
Hundreds of demonstrators turned up at the former Haslar naval base in Gosport.
Labour's deputy leader was speaking to BBC local radio ahead of the party's annual conference
Three people are charged after a protest in Gosport sparked by reports of an arrival of migrants.
