Inquest, a charity that provides expertise on state-related deaths, has reportedly submitted detailed concerns over the shooting to the UN human rights office according to The Observer.
In the same correspondence, the charity also voiced its displeasure at the recent death of Oladeji Omishore.
Mr Omishore, 41, died after plunging from Chelsea Bridge after being Tasered by officers in June.
Both cases sparked a huge public outcry and allegations of potential police mishandling.
The Kaba and Omishore families want the cases to be considered by the UN’s Office of the High Commissioner for Human Rights, which has examined “police brutality” and human rights “violations by law enforcement against Africans and people of African descent”.
Deborah Coles, director of Inquest, said: “There are legitimate concerns about the immediate resort by police to use of lethal force and racial stereotyping that equates black men with dangerousness and criminality.”
A statement from the Omishore family read: “We are extremely frustrated with the decision by the Independent Office for Police Conduct to treat the officers involved as witnesses and not declare a conduct or criminal investigation for what we saw as a disproportionate and excessive use of force. We strongly believe this force led to the untimely and avoidable death of Deji.
“There was no apparent risk assessment or efforts to de-escalate the situation in accordance with police guidelines. Deji was clearly in distress and experiencing a mental health crisis. We strongly believe the officers’ actions were unlawful, and a very sad reminder of how black people are treated in police contact.”
Mr Kaba’s mother Helen Nkama said it had been "very hard" to watch the footage and wants "justice".
A homicide probe by the Independent Office for Police Conduct is under way and the officer involved has been suspended.
The IOPC said: “We will examine whether Mr Kaba’s race influenced any actions taken by police.”