"Active U.S. support for the drug war in Mexico has been the status quo for more than a decade-it has fueled one of the deadliest conflicts in the western hemisphere and destabilized whole regions of the country", said Devereaux.
Five children injured in the attack were transferred to hospitals in the United States, U.S. and Mexican authorities said.
The LeBarons are descendants of Mormons who moved to Mexico in 1924, after disagreeing with the central church over polygamy. "Though it is our understanding that they are not members of The Church of Jesus Christ of Latter-day Saints, our love, prayers and sympathies are with them as they mourn and remember their loved ones".
All of the dead and wounded had dual US and Mexican citizenship, family members said.
In 2009, a prominent member of the clan, Benjamin LeBaron, 31, was shot dead near his community in northern Mexico.
Jhon LeBarón, a relative of the victims, said the 17 family members were following each other in a three-vehicle convoy when the ambush occurred.
Mr LeBaron named Maria Rhonita Miller and four of her children, including twins, as among the victims. "My heart is completely broken", he wrote.
LeBaron had broken down in her SUV when gunmen opened fire, torching the vehicle and causing the petrol tank to explode. "They left 6 of Dawna's kids on the side of the road and took off".
In another vehicle, 43-year-old mother Dawna Ray Langford was driving.
Mexican President Andrés Manuel López Obrador said he would speak with Trump about possible cooperation on security, provided Mexico's sovereignty was upheld.
In a tweet posted overnight, USA ambassador to Mexico, Christopher Landau, said he was keeping a close eye on the situation. He also seemed to rebuff Trump's offer of military help to fight the perpetrators, saying his government will seek justice in the case.
Mr. Trump said "a wonderful family and friends from Utah got caught between two vicious drug cartels, who were shooting at each other, with the result being many great American people killed, including young children, and some missing".
A few minutes later, he added: "This is the time for Mexico, with the help of the United States, to wage WAR on the drug cartels and wipe them off the face of the earth. We merely await a call".
Later, the two leaders spoke by telephone, and Trump offered US assistance "to ensure the perpetrators face justice", the White House said without giving details.
The U.S. Embassy and Mexico's federal Department of Security and Citizens' Protection did not immediately respond to requests for comment.
"The security of our nationals is our top priority".
Authorities have not publicly identified all of the victims or the circumstances of their deaths.
In each of the three cars attacked, a mother was driving children to see family.
One woman was killed after she apparently jumped out of her vehicle and waved her hands to show she wasn't a threat, according to family members and prosecutors. "Mothers were screaming for the fire to stop", he said.
Six other children were found alive. Some of the survivors are seriously wounded, including a 9-month-old who was shot in the chest and a 4-year-old shot in the back.