Updated

Iraqi Kurdistan has slapped a limit of 3,000 a day on refugees entering from Syria, after more than 30,000 flooded in from their war-torn homeland in a matter of days, aid agencies said Tuesday.

A new wave of Syrian Kurds began pouring into northern Iraq last week, seeking refuge from fighting between Kurdish forces and Islamist rebels, and a collapsing economy.

Jumbe Omari Jumbe, spokesman of the International Organization for Migration, said the authorities in Iraq's autonomous Kurdistan region had imposed a daily quota although it has not been applied to the letter.

"Today, they will allow 3,000 persons in, though yesterday a similar quota was also set, of 3,000, but at the end of the day, 5,000 refugees were allowed to cross," he told reporters in Geneva.

Jumbe said the refugees arriving at two border crossings in the region were exhausted and dehydrated after walking long distances in spiralling desert temperatures.

Access to Iraq has been erratic for Syrians, with political tensions and fears of a spillover of the conflict leading the Iraqi Kurdish authorities to shut the border in May.

Restrictions were eased last month to allow Syrians to join family members already in Iraq, but the number allowed to cross the border had remained relatively low.

Last Thursday, however, the authorities opened a crossing over a bridge at Peshkhabour on the Tigris River.

The influx began right away, with women, children and the elderly making up the bulk of those crossing.

On Saturday, Kurdistan closed the bridge to all but goods traffic, and opened another crossing for the refugees further south at Sahela.

Aid agencies have scrambled to get emergency supplies to the border zone and transport the refugees to camps deeper inside Iraqi Kurdistan.

UN refugee spokesman Dan McNorton said between 2,000 and 3,000 people were thought to be waiting to cross Tuesday, but that the numbers on their way were unclear.

"This is a situation where we are seeing a huge exodus of people fleeing the conflict in Syria," he added.

Over 1.9 million Syrians have fled their homeland since the revolt against President Bashar al-Assad erupted in 2011.

Most have found a haven in Lebanon, Jordan and Turkey, while UN figures put the number in Iraq at at least 185,000, including the latest influx.