The Independent Police Complaints Commission (IPCC) has launched an investigation after Nazim Din, 54, died during a police operation in Wortley, Leeds, on Wednesday night.
It is understood that Mr Din was trying to explain to officers they had arrested the wrong son when he suffered a suspected heart attack.
There were further claims the grandfather had been pushed by an officer – a claim West Yorkshire Police has refused to comment on while the investigation continues.
It was also claimed that emergency services had been held up by a large presence of up to 20 police cars and 45 officers.
A statement on behalf of Mr Din’s family said: “Nazim had much to live for and did not deserve to die in the way that he did. He had done nothing wrong.
“We urge the police to fully co-operate with those investigating Nazim’s death so that the facts can be brought to light as quickly as possible.”
A West Yorkshire Police spokesman said officers had gone to the address “as part of pre-planned arrest enquiries” and the force was “co-operating fully” with the IPCC.