McAleenan has been overseeing the USA southern border where more than 7,000 migrants have gathered near Tijuana, Mexico.
Applying for asylum at a USA land border can take months, so if migrants enter illegally and present themselves to authorities, their cases could be heard more quickly.
The migrants in Tijuana are part of several caravans that traveled through Mexico in an effort to enter the United States, citing issues such as widespread violence and dismal job prospects in their home countries.
The suspect who helped the children over the border wall to the group did not cross into the U.S., CPB said, according to Yuma's KYMA-TV. Another three managed to squeeze themselves through a narrow space in the fence.
Karen Mayeni, a 29-year-old Honduran mother with three children aged between 6 and 12, told Reuters that she was standing by and watching others penetrating the border and "waiting to see what happens".
One of the children "suffered a facial injury" and was treated on the scene. We've got well over 7,000 migrants there. The woman said she would decide her family's next action "in a couple of days". Mexican authorities arrested and deported those they considered responsible for compromising the city's commitment to public order.
"They were brought to the border by a group that told them they would be able to cross easily into the U.S.to present asylum claims, and that's not the case".
"We've got a challenging and still potentially volatile situation in Tijuana", said Kevin K. McAleenan, the commissioner of U.S. Customs and Border Protection, on Fox News Tuesday morning.
U.S. officials have restricted applications through the Chaparral gate in Tijuana to between 40 and 100 per day.
"Since that caravan formed in mid-October, we've seen 90,000 people come to our border, 85 percent of those crossing illegally between ports of entry, and all lured by the fact that our legal framework has huge gaps that create the opportunity to stay in the USA while awaiting your court hearing, even if they don't have a lawful permission or protection claim".
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