Syria hosts half a million Palestinian refugees, descendants of those admitted after the creation of Israel in 1948, and has always cast itself as a champion of the Palestinian struggle, sponsoring several guerrilla factions.
The rebels, fighting to topple President Bashar Assad, took full control of Yarmouk camp on Monday, rebel and Palestinian sources said, and government forces have been shelling the camp which is 2 miles from the city center.
The battle in Yarmouk is one of a series of conflicts on the southern fringes of Assad's capital, as rebels try to choke off the power of the 47-year-old leader after a 21-month uprising in which 40,000 people have been killed.
Abu Ali, 75, said he left his home in Yarmouk on Tuesday morning with his wife and three children as artillery shells rained down on the neighborhood of densely built apartment blocks.
"We walked out on foot without our belongings until we reached central Damascus. We got in a taxi and drove straight for the border," said the elderly man at the Lebanese Masnaa border post.
Abu Ali says he will stay with relatives in Ain al-Hilweh, a Palestinian refugee camp in Lebanon's southern city of Sidon. He estimated that 70 percent of Yarmouk's residents had fled but added that many slept rough on the streets on Damascus.
More than half a million Syrian refugees have fled into neighboring countries during the revolt, and the UN refugee agency say 3,000 are now seeking refuge abroad daily.
Lebanon -- the closest country to Damascus -- hosts 154,000 but aid agencies say a rebel push into the Syrian capital could force more than 10,000 to flee in hours.Destruction
Assad's government and the mainly Sunni Muslim Syrian rebels have both enlisted and armed Palestinians as the uprising has developed into a civil war.
The battle has pitted rebels, backed by some Palestinians, against Palestinian fighters of the pro-Assad Popular Front for the Liberation of Palestine-General Command. Many PFLP-GC fighters defected to the rebel side and their leader Ahmed Jibril left the camp three days ago, rebel sources said.
Aid workers said on condition of anonymity that street clashes had reduced in intensity on Tuesday and there was no sign of the PLFL-GC in Yarmouk but government bombardment continued and some fleeing residents were sitting in parks in the neighboring districts of Midan and Zahra.
They said rebels were spread out across the camp, including in the abandoned PFLP-GC headquarters.
Um Mohammed, 65, arrived at the border in a bus with several Palestinian families. "There is a huge amount of destruction. There are armed men in the street," she said.
Others said that rebels, supported by the Palestinian fighters, were taking control of the area.
"The camps are 80 percent controlled by the (rebel) Free Syrian Army and Palestinian fighters," said Muna, a young woman who entered Lebanon on Tuesday afternoon. "All of the (pro-Assad) PFLP fighters have left the area.
It is not clear if the Syrian army will try to reenter Yarmouk in the coming days but Muna said she could see troops massing as she left the area.
"But the army is on the outskirts of Yarmouk. We passed several government checkpoints to get out."
MASNAA, Lebanon (Reuters) -- More than 1,000 Palestinian refugees living in Syria have crossed into Lebanon in the past 24 hours, a source at the Lebanese border said on Tuesday, after Syrian rebels took control of a Palestinian refugee camp in Damascus.