JERUSALEM — United Nations chief Ban Ki-moon heads to the Middle East today to prod efforts to end fighting between Israel and Gaza Strip militants that has pushed the Palestinian death toll past 300 and sent millions of Israelis running for cover from rocket fire.

As Israel pressed ahead with the ground offensive it launched on July 17, artillery fire killed eight members of one family and five members of another, the Gaza Health Ministry said in a statement. Soldiers, backed by tanks, artillery, aircraft and warships, moved into the Hamas-controlled enclave after 2,100 air strikes over 10 days and Egyptian truce efforts failed to quell rocket barrages.

Military spokesman Lt. Col. Peter Lerner said by phone that the militants’ arsenal of 10,000 rockets has been diminished by almost half since the campaign began. Ground troops are demolishing some of the dozens of tunnels into Israel that militants built to stage raids, he said, the declared objective of the incursion. Another cross-border raid by militants who entered Israel through a tunnel was foiled today, the military spokesman’s office said.

Israel and Gaza militants have clashed frequently since the Israeli government evacuated soldiers and settlers in 2005, while limiting the movement of people and goods in and out of the territory by maintaining control of border crossings. The ground campaign is Israel’s first serious incursion into the territory since 2009. Lerner described it as open-ended. 

Ban will travel to the Middle East to consult with regional leaders on the Gaza situation, Jeffrey Feltman, the U.N. under- secretary-general for political affairs, told reporters in New York yesterday. The U.N. chief has denounced the rocket fire while calling on Israel to “do far more” to stop civilian casualties.

The Palestinian dead include dozens of children, according to Gaza Health Ministry official Ashraf al-Qedra. Three Israelis have also been killed, including a Bedouin man today.

Mohamed al-Fayoomi, a 47-year-old father of eight, said he fled his home in Gaza City early yesterday after artillery shells fell nearby. “I took my wife and children,” he said. “I found hundreds of people had also left their homes in the darkness.”

Al-Fayoomi said he went to stay with his sister in a western part of the city. Others have moved to the south, according to The United Nations Relief and Works Agency, and more than 40,000 are sheltered in U.N. facilities in northern and central Gaza.

Irael’s chief military spokesman, Brig. Gen. Moti Almoz, told Israel Radio that about 50,000 Gazans left their homes after receiving evacuation warnings from the military, and that additional notices were to go out later today.

Israeli Prime Minister Benjamin Netanyahu said Friday that he instructed the military to prepare for a “significant expansion” of the ground offensive. The government has authorized the call-up of 65,000 reservists, and more than 55,000 have been mobilized so far, military spokesman Capt. Eytan Buchman said by phone.

Ground troops poured into the densely populated coastal area after Israel reported foiling a raid by 13 Gaza militants who crossed the border through an underground tunnel. Lerner said soldiers have taken over 13 tunnels so far.

“We have substantially struck the tunnel infrastructure of Hamas,” he said. While Israel has been aware that militants have been digging a tunnel network over the past two years, operations against them have been constrained in quieter times, he said.

Troops so far have taken up positions on the periphery of built-up areas as they deal with the tunnels, he said. Risks they’ve incurred include at least one explosives-strapped donkey that exploded at a safe distance after forces engaged it, according to an e-mailed statement from the military.

Israel has also destroyed 3,000 rockets and 1,100 rocket launchers, Lerner said. Taking into account the more than 1,600 rockets militants have fired, “they’re down about 50 percent of their force,” he said.

While rocket fire has dropped off in recent days, it’s premature to declare a trend, Lerner said. Most of Israel’s 8 million people live within the range of Gaza rocket squads, and missile defenses intercepted 346 headed for built-up areas, according to the military’s tally.

Hamas and the Islamic Jihad group rejected Egypt’s truce proposal earlier this week after Israel accepted it, saying it didn’t guarantee lifting the blockade on the coastal enclave that has resulted from Israeli and Egyptian control of border crossings.

A provision in the proposal said Gaza’s passages would be opened for both people and goods “once the security situation becomes stable on the ground.” Egyptian Foreign Minister Sameh Shoukry said today that Egypt has no plans to revise the cease- fire proposal.

Palestinian President Mahmoud Abbas, who has been critical of the Hamas rocket fire, has met this week with Egyptian and Turkish leaders trying to find a formula to end the violence. French Foreign Minister Laurent Fabius, who backed Egypt’s truce call in Cairo today, is headed to Israel today.

The violence flared after youths on both sides of the conflict were kidnapped and killed in recent weeks, and at a time when Hamas is economically pressed by the loss of its patron in Egypt, ousted Islamist President Mohammed Morsi. The U.S. and European Union label Hamas a terrorist organization.