Ukraine has accused Moscow of an "invasion" after a Russian aid convoy crossed its eastern border without permission.
But Kyiv said it would allow the trucks safe passage in order to avoid escalation of ongoing conflict.
Some 145 Russian trucks from a group nearly twice that size that had been parked near the border for a week crossed into Ukraine on August 22, after Moscow accused Kyiv of deliberately causing delays.
Ukraine has voiced concerns the convoy could supply pro-Russian rebels for their fight against government forces struggling to regain control of the Luhansk and Donetsk provinces, or place the trucks strategically to hamper Ukrainian forces' efforts.
Ukrainian President Petro Poroshenko called the convoy's entry a "flagrant violation of international law." In comments on the presidential website, he said more than 100 trucks had crossed the border without checks by Ukrainian customs or border guards and unaccompanied by International Committee of the Red Cross (ICRC) officials.
Russian UN Ambassador Vitali Churkin, speaking in New York, said that Russia hopes the Red Cross told Moscow they might participate in the delivery of the aid "because there are some ICRC people on the ground [in Luhansk]."
But Ukrainian Deputy UN Ambassador Oleksandr Pavlichenko that Ukrainian customs officials who had started clearance of the convoy the previous day, on August 22, "were blocked by the Russian forces and prevented from the inspection of the rest of the trucks in the convoy."
Pavlichenko said that as a result, neither Ukraine nor the Red Cross are aware of the contents of the cargo of the Russian trucks, which "entered the territory of Ukraine without proper border and customs procedures."
NATO Points Finger At Russian Troops
A Pentagon spokesman said the U.S. government "strongly condemns" the unilateral movement of the Russian convoy into Ukraine.
Rear Admiral John Kirby, the Pentagon press secretary, called on the Russians to immediately remove the convoy.
White House adviser Ben Rhodes also called on Russia to withdraw the trucks from Ukraine or "face consequences."
NATO and the European Union were also quick to condemn the Russian move without the consent of the Ukrainian authorities and without any involvement of the ICRC.
NATO Secretary-General Anders Fogh Rasmussen called it "a blatant breach of Russia's international commitments."
Rasmussen also said on the alliance had observed an alarming build-up of Russian ground and air forces in the vicinity of Ukraine. He said Russia was fueling the conflict with "artillery support -- both cross-border and from within Ukraine."
NATO Supreme Commander U.S. General Philip Breedlove said, "A forced crossing without authorization or escort indicates that Russia is more interested in resupplying separatists rather than supporting local populations."
European Union foreign policy chief Catherine Ashton said she deplored Russia's decision for the convoy to enter Ukraine without a Red Cross escort and without Kyiv's consent, according to her spokesman. She urged Russia to reverse the decision.
Late on August 22, Lithuanian Foreign Minister Linas Linkevicius announced that his country's honorary consul in Luhansk had been "kidnapped and brutally killed by terrorists there," using the same term for the perpetrators that the government in Kyiv uses for pro-Russian separatists.
The first trucks rolled into the rebel-held city of Luhansk hours after they crossed the border, local officials and accompanying journalists reported.
Ukrainian Security Service chief Valentyn Nalivaychenko called the trucks' entry a "direct invasion." But he said Kyiv would not use force in order to avoid "provocations," and the Foreign Ministry echoed that message.
"In order to avoid provocations we gave all the necessary commands for the safe passage of the convoy," the ministry said. It urged other nations to condemn what it called "Russia's illegal and aggressive actions."
Putin Defends Move
The Kremlin said President Vladimir Putin told German Chancellor Angela Merkel in a phone call after the trucks rumbled over the border that further delays in delivering the aid would have been "unacceptable."
Putin told Chancellor Angela Merkel that after "clear procrastination" by Ukraine, Russia had decided to send the convoy to its intended destination.
Russian Deputy Foreign Minister Sergei Ryabkov released a statement saying Russia had acted within the boundaries of international law and added, "We cannot and will no longer put up with the misery of people living in…southeastern Ukraine."
Russian news agency ITAR-TASS quoted a Kremlin spokesman as saying the chiefs of staff of Poroshenko and Putin had agreed on the need for the swift completion of the convoy's operations.
The Red Cross, whose participation was meant to assuage Kyiv's concerns about the purpose of the convoy, said it was not escorting the trucks due to the "volatile security situation."
ICRC spokeswoman Galina Balzamova said no "adequate security guarantees" were given by the fighting parties, so the Red Cross "is not taking part in this convoy and is not accompanying it."
The Red Cross also said that its team in Luhansk had reported heavy shelling overnight.
The convoy of more than 250 white trucks left the Moscow region on August 12, carrying what Russia said was more than 2,000 tons of aid for residents of Luhansk.
Trucks began to cross the border shortly after Moscow said it was fed up with delays it blamed on Ukraine.
"All excuses to delay the delivery of aid...have been exhausted," the Russian Foreign Ministry said in a statement.
"The Russian side has decided to take action. Our convoy carrying a humanitarian cargo is starting to move in the direction of Luhansk," it said.
It said the trucks had long been ready for inspection by Ukrainian border and customs officers but that Kyiv had repeatedly come up with new demands.
The United Nations says more than 2,000 people have been killed in months of fighting in eastern Ukraine following the ouster of a Ukrainian president sympathetic to Russia in February and Russia's annexation of Crimea in March.
The fighting has prompted thousands of people to flee and living conditions have deteriorated. The city of Luhansk has been without fresh water or regular power supplies for days.
Fighting has persisted in eastern Ukraine as Moscow and Kyiv wrangled over the convoy, and many of the deaths have occurred in the past month.
The cities of Luhansk and Donetsk remain in rebel hands despite recent gains by government forces.
Russia warned against any attempts to "undermine the humanitarian mission." It said it was ready for the Red Cross to escort the convoy and help distribute the aid.
"We declared once again: All the necessary security guarantees...have been provided," the Russian statement said.
"Such excesses, candid lies, and inability to reach agreement can be tolerated no more," Russia said.
ITAR-TASS quoted a Russian customs official as saying Russian and Ukrainian customs officers had jointly completed customs formalities for 34 of the vehicles. The ICRC tweeted on August 21 that Ukrainian border guards had started inspections of some of the Russian trucks.
Fighting, sometimes fierce, has persisted in eastern Ukraine as Moscow and Kyiv wrangled over the convoy. Many of the deaths have occurred in the past month, and the cities of Luhansk and Donetsk remain in rebel hands despite gains by Ukrainian governent forces in recent weeks.
Moscow's annexation of Crimea and the conflict in eastern Ukraine have driven ties between Russia and the West to post-Cold War lows and prompted exchanges of economic sanctions, visa bans, and asset freezes.
The ICRC's director of operations, Dominik Still Resin, said that organization was "committed to work with all sides to bring aid to people in east #Ukraine in a neutral and independent manner."
With reporting by Reuters, AP, AFP, dpa, ITAR-TASS, and Interfax