The four -- two Kenyans, an Indian and a Pakistani -- were arrested last week with 98 packets of suspected heroin, and were subject to an Interpol "red notice" from the US for their arrest.
"Kenya is under a duty to cooperate because international crime has hurt Kenya a lot," magistrate Maxwell Gicheru told the court in the port city of Mombasa.
The four, who were remanded in custody while police process the extradition request, are due in court again on December 1.
Their wealthy father, Ibrahim Abdalla Akasha, was gunned down in Amsterdam in 2000, after fleeing Kenya where he was sought in connection with the seizure of several tonnes of hashish.
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The two other suspects are Vijaygiri Anandgiri Goswami, an Indian, and Kulam Hussein, from Pakistan.
With drug smuggling routes from Asia through the Middle East hampered by conflict and tougher border restrictions, traffickers have increasingly turned to east Africa as a transport hub, according to UN Office on Drugs and Crime (UNODOC).
In April, British and Australian navies seized the largest ever haul of heroin at sea, weighing 1,032 kilogrammes.
The drugs, worth some USD 235 million, were found on a dhow some 50 kilometres off the coast of Kenya and Tanzania.
East Africa has a small but growing domestic market for heroin, but the vast majority shipped to Kenya and Tanzania is believed to be then moved onwards to South Africa and west Africa, and potentially on to Europe.