The flow of thousands of refugees began on Sunday after Venezuela's National Bolivarian Armed Forces launched an offensive against illegal armed groups in La Victoria, a municipality in Venezuela's Apure border province.
Those fleeing accused members of the Venezuelan military of committing abuses, including detaining and killing civilians, as well as looting and burning homes.
"They raided our house and took everything from us. When they arrived they broke everything, the doors, they entered and took everything I had in the house, the workshop," mechanic Jose Castillo, who arrived in Colombia with his pregnant wife and 12-year-old daughter on Friday, told Reuters.
"I couldn't stay because they are killing people. They killed some neighbors and dressed them in Venezuelan army uniforms to pass them off as guerrillas," Castillo said.
Reuters could not independently verify Castillo's accusations, or those of other displaced Venezuelans who showed photos on their cell phones of dead people wearing camouflaged uniforms with weapons next to their hands.
The victims were residents of La Victoria and its surroundings, they said.
Venezuela will investigate events in La Victoria, the country's chief prosecutor Tarek Saab wrote in a Twitter message on Friday, and punish those responsible if abuses of human rights are discovered.
Hundreds of children, women, elderly, and men are staying on mattresses and in makeshift tents, set up in sports centers in Arauquita, in Colombia's Arauca province, where they have access to food and sanitation.
Earlier this week two Venezuelan soldiers were killed in clashes in Apure, as was the leader of an illegal armed group, while 32 people were captured, the Venezuelan government said on Monday.
Dissidents of the demobilized Revolutionary Armed Forces of Colombia (FARC), who reject a 2016 peace deal with the Colombian government, are the targets of the military operations, according to the civilians fleeing across the river that divides Colombia and Venezuela.
Some 4,000 people have arrived in Colombia from Venezuela since Sunday due to the clashes, Juan Carlos Agudelo, government secretary of Arauquita, told Reuters.
"The situation is confusing, delicate," he said.Colombian President Ivan Duque has accused the government of his Venezuelan counterpart Nicolas Maduro of sheltering FARC dissidents and members of the National Liberation Army (ELN) in Venezuela, something the government in Caracas denies.