Peru has brought into force new entry requirements to control the numbers of Venezuelan migrants.
Venezuelans will no longer be admitted with just an identity card, although children, pregnant women and the elderly are exempted.
Similar rules were introduced in Ecuador last week, only to be overturned by a court.
Tens of thousands of Venezuelans are fleeing their country amid chronic shortages of food and medicines.
The country’s longstanding economic crisis has seen more than two million citizens leave since 2014, causing regional tensions as neighbouring countries struggle to accommodate them.
The UN is setting up a special team to co-ordinate the regional response.
However, the UN’s migration agency has warned that the continent faces a refugee “crisis moment” similar to that seen in the Mediterranean in 2015.
The authorities in Lima set a deadline of midnight on Friday for the new passport rules to come into force.
Many Venezuelans have been looking to start a new life in Peru – which has one of the region’s fastest growing economies – travelling there via Colombia and Ecuador.
On Friday there was a rush of people heading to the Peruvian border and Ecuador created a “humanitarian corridor” by laying on buses to take migrants through the country.
On Friday, more than 2,500 crossed into the small Peruvian border town of Aguas Verdes, with thousands more trying to reach Peru at the main crossing point at Tumbes.
Many were exhausted by their journey.
“We have been on the road for five days. We travelled by bus and saw people, Venezuelans, walking along the road,” Jonathan Zambrano, 18, told AFP news agency.
The Tumbes crossing has seen about 3,000 arrivals per day in recent weeks.
Peru is already home to about 400,000 Venezuelan migrants, most of whom arrived in the past year.
Joel Millman, a spokesman for the International Organization for Migration (IOM) – a UN agency – said increasing border restrictions coupled with an outbreak of violence on the Brazilian border last week, 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. (Excerpts from BBC)