CAIRO—Thousands of residents of Sudan’s oil-rich disputed Abyei region continued to flee south on Tuesday, along with humanitarian groups attempting to prepare clinics and shelters before the rainy season strands the displaced and renders mostly dirt roads impassable.

Chol Anguie, a member the Abyei administrative council, said hundreds of children were separated from their parents when fighting started in the region over the weekend and are now searching for their families in towns to the south. Abyei had an estimated population of about 20,000.

“Their situation is very bad and they need urgent assistance,” Chol said.

Some of the same families were separated from their children during the country’s civil war, others said.

“We are doing our best not to lose our children again. Some were lost these past three days,” and some may have been wounded in the fighting and died, said Koul Deng, paramount chief of the Dinka Ngko tribe, reached on Tuesday in Agok, a town to the south of Abyei.

He did not have figures for those wounded or killed, he said.

The mostly Christian and animist south and Muslim north fought a long civil war that took more than 2 million lives before a 2005 peace treaty. The south voted in January to form a separate nation this summer but both regions of Sudan still lay claim to Abyei. Deng predicted the battle for control of Abyei would rage on in coming days.