Ukrainian troops besieged by pro-Russian separatists abandoned two military bases in the embattled east on Wednesday, as the incoming president promised to announce a peace plan soon after his inauguration this weekend.

The humiliating retreats underscored the Ukrainian military’s weakness as it wages an offensive against separatists fortified by armaments brought across the border from Russia.

After 10 hours of fighting at a building housing a Ukrainian national guard regiment near the city of Luhansk, soldiers ran out of ammunition, the national guard said. The battle began when more than 300 heavily armed insurgents opened fire on the base from positions in residential apartment buildings and on rooftops, according to the statement. Six militants were killed, three Ukrainian soldiers were injured and the rebels eventually took control of the building, the statement said.

Separatists also overran a border guard base just outside Luhansk after two days of fierce fighting. The border service said the troops on the base had been “redeployed.” But a YouTube video suggested a more ignominious defeat, showing Ukrainian troops in civilian clothes filing out past watchful rebels and carrying tote bags containing their belongings.

Many Ukrainians say their military is weak because its resources were systematically looted during the presidency of Viktor Yanukovych, who was ousted from office in February after months of protests. The military is so starved for supplies that Ukrainian citizens have raised money to buy ammunition, food and other provisions for it.

The United States has said that it will send Kiev an additional $5 million in equipment, including body armor, night-vision goggles and communications gear. Until now, the assistance has been limited to nonlethal supplies, such as meals-ready-to-eat and tents. Meanwhile, Ukraine’s military continues to suffer casualties in daily clashes with insurgents, some of whom are Russians who have crossed the border to assist in the fighting.

Combat in the industrial east has intensified ahead of the inauguration Saturday of Petro Poroshenko, a chocolate magnate who was elected May 25. The government aims to bolster the border with Russia and stop the influx of fighters and supplies, but its efforts have been ineffectual. Thousands of residents have fled their homes.

Interim President Oleksandr Turchynov has asked Ukraine’s Security and Defense Council to discuss imposing martial law in the east to calm the region.

Poroshenko met Wednesday with President Obama in Warsaw and said he will waste no time proposing his peace plan, vowing that “from the very beginning, from the first day of inauguration, we are ready to present the plan for peaceful relations [regarding] the situation in the east.”

Obama said after the talks that Ukraine can become a thriving democracy if the international community stands behind it. He pledged that the United States would provide new support as Ukraine’s government seeks a way out of the crisis.

“The Ukrainian people made a wise selection in somebody who has the ability to lead them through this difficult period,” he said, referring to Poroshenko.

He said Ukraine’s elections last month represented an “incredible outpouring of democracy in the face of actions by Russia as well as armed militias in certain portions of the east that violate international law, violate sovereignty and have spurred great violence.”

What Ukrainians made clear in the elections “is that they reject that past,” Obama said. “They reject violence. They reject corruption.”

Saying he is “deeply impressed” by Poroshenko’s vision for Ukraine, Obama called on the world “to stand solidly behind the efforts of Petro to broker with the Russians a process whereby Russia no longer is financing or supporting or arming separatists on Ukraine’s sovereign territory.”

The United States and Europe accuse Russia of fomenting the unrest after it annexed the autonomous Ukrainian region of Crimea in March.

In Brussels, NATO’s senior military leader said Wednesday that the majority of the up to 40,000 troops Russia had deployed along the Ukrainian border have been withdrawn or are preparing to leave. But he said that “several large formations . . . look like they intend to remain.”

Overall, Russia continues efforts to “destabilize” eastern Ukraine by promoting “very well-led, very well-financed, very well-organized clashes with Ukrainian forces,” said Gen. Philip M. Breedlove, NATO’s supreme allied commander. “It is very clear that the Russian influence is a part of this.”

“Russian irregular forces, Russian-backed forces and Russian finances are very active in eastern Ukraine,” he said. He declined to identify the “irregular” forces but noted reports that fighters from Chechnya had been previously identified among separatist groups in Ukraine.

In a meeting of NATO defense ministers, Breedlove outlined the alliance’s response to the crisis in Ukraine and proposals for medium- and long-term changes to NATO’s force posture in Europe.

“Over the past 10 to 12 years, NATO and individual nations and leadership have made decisions about force structure, readiness and provisioning . . . based on seeing Russia as a partner,” he said. “That has changed. We now see a much more aggressive neighbor who is willing to amass large forces on the border of another nation, bring that force to very high readiness, to deep provisioning and supply, and at least in one case cross an internationally recognized border and annex part of a sovereign nation.” He added, “That changes how we have to plan and arrange our forces in NATO.”

Defense chiefs in Brussels also met with their Ukrainian counterparts, who presented a new list of desired military assistance. “The Ukrainians have asked for all manner of aid . . . everything from command and control, communications, help with cyber, help with their kinetic capability,” Breedlove said.

Deane reported from London. Karen DeYoung in Brussels and William Branigin in Washington contributed to this report.