But while some are giving up their dream, others are just arriving, further stretching resources at makeshift camps.
Many migrants trying to enter the United States are fleeing violence and poverty in Central America, but they have been seized on by President Donald Trump, other Republicans and conservative news outlets as a national security threat.
The President later also denied that the tear gas was used against children.
But as migrants reached the border area, some protesters split off toward multiple locations, CBP said, attempting to gain entry to the USA through the port and "directly east and west" of it.
He also said the tear gas was "very safe" and the people suffering from its effects "to a certain extent were the people that were putting it out there".
Following the weekend skirmish - which reportedly led to more than 100 combined arrests on both sides of the border - Mexico's Interior Ministry announced Monday it planned to deport 98 migrants. He questioned why parents were approaching the border with children at all, suggesting that some of the adults "grabbed" children to better their chances of entering the US.
It was not clear from the ministry's statement how numerous migrants were among those arrested on Sunday or when the deportations would take place.
"Mexico's Milenio television station showed dozens of people climbing the fence on the Mexican side of the border before they were turned back".
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White House spokesman Hogan Gidley said Trump and GOP leaders discussed a range of legislative priorities, including the California wildfires, the farm bill and "the great need for border security at this crucial time", as well as prison reform and the budget.
Some have chosen to accept temporary work and asylum from Mexico, Fox News said.
Trump has deployed almost 6,000 troops to the border, on top of more than 2,000 members of the National Guard, but Scott said the Department of Defense was not involved in Sunday's engagement.
Mina and his wife and their toddler daughter avoided the march and were glad they did after hearing others recount what unfolded, he said as he sat in the doorway of his family's tent at Tijuana sports complex using a toothbrush to clean the fine dust that coats everything off his sneakers. Five million US -born children with American citizenship are said to be living with parents or relatives who are unauthorized.
US lawmakers must act to pass a spending bill by December 7 to fund some government agencies, including the Department of Homeland Security that oversees the border and immigration.
Border agents were forced into action because they were "being rushed by some very tough people", Mr Trump said.
"Entering our country illegally, particularly over our walls is not only risky, but also very foolish", San Diego Sector Chief Patrol Agent Rodney Scott said in a statement.
Of the 6,062 total in Tijuana, according to officials, 3,877 men are men, 1,127 are women and 1,058 are children. Mexican authorities said 39 people were arrested in Mexico. The chaos followed what began as a peaceful march to appeal for the USA to speed processing of asylum claims. "Mexico wants to see if they can get it straightened out".
Four agents were hit with rocks, but due to their protective gear they were not seriously injured, McAleenan said.