Thousands of Christian refugees fled fighting in southern Lebanon today, many of them toward Israeli lines, as Druze and Moslem militias pressed an attack on one of the few remaining Christian strongholds in the region.

The heaviest fighting was reported around the village of Kfar Falous, the last major barrier between the attacking forces and Jezzin, a Christian city of about 15,000 people seven miles to the east.

The Druze and Moslem forces have taken advantage of the phased pullback of Israeli troops, under whose protection the Christians expanded their territorial control, to drive out the Christians, their enemy in a 10-year-old on-again, off-again civil war.

Beirut radio said the Druze and Moslem units had entered the village and that fighting was concentrated around a nearby Lebanese Army base, where Chritian militiamen were deployed. The Christian Voice of Lebanon radio, however, said Christian forces had repelled a major offensive on the base today and reporters at the scene said the Christians remained in control of Kfar Falous.

Druze leader Walid Jumblatt pledged not to attack Jezzin, The Associated Press reported. "I will not attack Jezzin and I will do my best not to allow it," he said at a news conference in his Chouf mountain headquarters. There was no clear indication of the Moslem forces' intentions.

The continuing fighting and growing pressure on Jezzin added to the flow of refugees headed toward Israel's so-called "security zone" along the Israeli-Lebanese border. Israeli Army units withdrew into the six- to 15-mile-wide security zone yesterday in the last step before their final pullout from Lebanon, which is expected in two or three weeks.

There was no accurate count of the number of refugees among the estimated 40,000 Christian residents of the areas that have already been overrun by the Druze and Moslem forces. Israeli military officials estimated that as of tonight 10,500 Christians had fled into the security zone, most of them seeking shelter in Marjayoun and nearby villages.

The officials said that among more than 2,000 refugees to reach the security zone today, about 40 had been wounded and eight were taken to hospitals in Israel. In addition, military officials here estimated that 20,000 to 30,000 Christians had fled from the west to Jezzin.

The Israelis have allowed a small number of Lebanese refugees to pass through Israel on their way to destinations abroad, but there appear to be no plans for a general opening of the border.

U.N. officials said about 3,000 refugees appeared to have reached a string of Christian villages east of Naqura, the coastal town that is the headquarters of the U.N. force in southern Lebanon, within the Israeli security zone.

U.N. officials said there were reports of long lines of waiting refugees at Kfar Houne, a major new crossing point from the north into the Israeli zone.

An Israeli radio reporter who toured the security zone today said village streets were jammed with refugees and their automobiles, piled high with belongings. The refugees were being housed in schools, churches and with relatives in the Marjayoun area, according to Israeli Army officials.

A woman from the Christian village of Miyumiye near Sidon told Israeli radio she had fled because "I have no more house there. It is burned, taken by the Palestinians. It seems that they are Lebanese and we are not Lebanese."

Christian residents of Miyumiye attacked and burned parts of the nearby Palestinian refugee camp earlier when the area was still under Israeli control.

Israeli officials said they were providing food, medicine and blankets to the refugees who reached the security zone and were arranging for the transport of similar nonmilitary supplies to Jezzin by the Israeli-backed South Lebanon Army. But the Israelis remained adamant in their opposition to any direct intervention in the fighting, telling the Christians in Jezzin that they were on their own.

Brig. Gen. Antoine Lahad, the Christian commander of the South Lebanon Army, has pledged to defend Jezzin against Druze or Moslem attack. Israel apparently is willing to give Lahad a free hand in such an endeavor but not to assist him. Moreover, since Israel long ago concluded that Lahad's force was incapable of standing on its own, it did not appear that an intervention by it would have a decisive effect on the fate of Jezzin.

An AP correspondent reported from near Kfar Falous that the South Lebanon Army was using Israeli-supplied Sherman tanks against the Druze and Moslem forces. There was no indication when it had received the tanks.

The flight of the refugees in the last few days is the latest in a string of disasters that has hit Lebanon's Christian community since the high point that followed Israel's June 1982 invasion of the country.

Long allied with the Israelis, the Christians expanded their territory and political influence following the invasion and saw the election of their leader, Bashir Gemayel, as president of Lebanon. But Gemayel was assassinated in September 1982 before he took office and was replaced by his brother, Amin, who has been a bitter disappointment to many of his fellow Christians.

Each step in Israel's pullout from Lebanon has been followed by serious setbacks for the Christians. When the Israelis evacuated the Chouf mountains southeast of Beirut, Druze militiamen overran most Christian villages in the area.

The Israelis withdrew from the Sidon area on Feb. 16, but the Christians were able to maintain their positions until last week, when the Israelis also pulled out of Jezzin and the surrounding area.

Since then, Druze forces in the mountains north of Sidon have swept west, overrunning a Christian area on their way to reaching the coast, while a combination of Druze, Moslem and Palestinian militiamen have driven the Christians from their villages east of Sidon back toward Jezzin.