The boy, identified by Texas congressman Joaquin Castro as Felipe Alonzo Gomez, had been transferred to a New Mexico medical center showing signs of sickness on Monday, the agency said Tuesday.
Hospital staff diagnosed the boy with a common cold but when he was evaluated for release they diagnosed a fever.
During transportation to the hospital, he began to vomit and lost consciousness.
With border crossings surging, CBP processes thousands of children - both alone and with their parents - every month.
The boy's death came at a time when the USA is undergoing a partial government shutdown, which was triggered by Trump's demand for border wall funding.
A timeline of events released by the agency shows the boy and his father were shuttled through three different processing facilities over six days - but CBP also said agents made more than two dozen "welfare checks" during that time, and the boy and his father had hot food, water and juice throughout.
"The cause of death of the minor is still being investigated and the medical records have been requested in order to help clarify the cause of death", the statement said.
Pro-migrants activists demonstrate next to US border patrol agents against US migration policies near the US-Mexico border fence at Imperial beach in San Diego County, US, as seen from Playas de Tijuana, Baja California state, Mexico, on 10 December, 2018.
US charges former Flynn business partner with secret lobbying for Turkey
Ekim Alptekin , a Turkish businessman regarded as close to Erdoğan, was also charged in the indictment , the filing shows. Gulen has been blamed by the Turkish government for orchestrating a failed military coup against Erdrogan's government.
On Tuesday, Castro called for a congressional investigation of the death. CBP officers and the Border Patrol remain on the job despite the shutdown.
"This is why I asked Congress on Tuesday to change our laws so that the United States is not incentivizing families to take this risky path", McAleenan said.
According to Guatemala's foreign ministry, the father and son entered the U.S.at El Paso, Texas, on December 18, then were taken to the Border Patrol's Alamogordo station Sunday.
In the wake of the boy's death, McAleenan announced a series of moves on Tuesday night.
The Guatemalan Foreign Ministry issues a statement that it was alerted to a minor's death in the US.
CBP said it apprehended Felipe and his father on December 18 about five kilometres away from an official port of entry, the Paso del Norte bridge connecting El Paso, Texas, and Juarez, Mexico.
Her body was repatriated on Sunday to Guatemala, and her remains were to be transported to the indigenous community of Raxruha, where she called home.
CBP typically doesn't keep immigrants in holding facilities for more than 72 hours because they don't have the services offered at the larger ICE detention centers. She was flown by helicopter to a hospital in El Paso, Texas, where she died.
The CBP said it had not established the cause of death but would "ensure an independent and thorough review of the circumstances". "Instead of immediately acting to keep children and all of us safe along our border, this administration forced a government shutdown over a wall".
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