The UN has warned Venezuela is heading for the same refugee "crisis moment" seen in the Mediterranean in 2015.
The warning comes as neighbouring countries try to stem the flow of Venezuelans looking to escape the country's dire economic situation.
Peru brings in stricter border regulations on Saturday - a day after a court overturned Ecuador's attempt to strengthen its own controls.
More than two million Venezuelans have fled their country since 2014.
They are fleeing a severe economic crisis which has led to shortages of food, medicine and basic goods, with many saying they are doing so because they cannot get the operations and medical care they need.
Now, the threat of becoming trapped has sent many more running for the border. Jonathan Zambrano, 18, told news agency AFP he had been on the road for five days, seeing many others walking towards Peru as he made the journey.
More than 2,500 crossed into a small Peruvian border town on Friday, according to the agency, with thousands more attempting to reach Peru at the main crossing in Ecuador.
They are trying to arrive before the new rule, requiring them to have valid passports, is brought into force on Saturday. Until now, Venezuelans have been allowed to enter Peru with just their ID cards.
Ecuador brought in a similar law last week. However, on Friday, a judge found requiring Venezuelans to carry valid passports broke regional agreements on freedom of movement.
The state of Roraima, in Brazil's Amazon region, also had its attempt to close the border thrown out by a judge earlier this month.
'It's hard to help'
A hardening towards Venezuelans seeking a new life could also be heard on the streets in Peru.
Giannella Jaramillo, who runs a clothes stall in a town near the border, told news agency AFP: "On the one hand, we're sorry for the Venezuelan people, but they are taking a job away from a Peruvian.
"It's hard to help more people."
Her words are echoed by Ecuadoran Gerardo Gutierrez.
"Walk two blocks and you see 10 Venezuelans, walk another two and you see 10 Venezuelans.
"In economically poor countries, it's hard to help more people with what little there is."
Joel Millman, a spokesman for the International Organization for Migration (IOM), pointed out all this - as well as recent violence at the border in Brazil - was an early warning sign that the region was in need of help.
"This is building to a crisis moment that we've seen in other parts of the world, particularly in the Mediterranean," he told reporters.
"A difficult situation can become a crisis situation very quickly and we have to be prepared," he added.
UN Secretary-General Antonio Guterres is to set up a special UN team which will co-ordinate a regional response to the crisis, while Ecuador is to host a 13-nation regional summit in September.
BBC
