Ukraine and Russia poured reinforcements into eastern Ukraine this weekend, preparing for what is likely to become the war’s biggest battles as refugees continued to flee the looming Russian assault.

Russia’s main objective now is to seize the parts of the eastern Donbas region not yet controlled by Moscow. Unlike the first phase of the six-week-old conflict, that shift is forcing Ukraine into fighting conventional battles involving tanks, artillery and aircraft on flat, often barren terrain that allows Russia to leverage its superiority in military equipment.

Fresh Russian tank and artillery units, as well as forces withdrawn from areas around Kyiv, began arriving in recent days to staging grounds for the offensive north of the Ukrainian city of Izyum, according to footage shown on Russian military television. Ukraine, too, started moving toward Donbas combat units from areas of northern Ukraine that it recovered after Russian troops retreated.

Ukrainian President Volodymyr Zelensky called for urgent assistance ahead of this new round of the conflict, warning that Moscow hasn’t given up on its aspirations to subjugate Ukraine. “Russia can still afford to live in illusions, gathering new armor and new troops on our soil. And that means that we need even more sanctions, and even more weapons for our state,” he said in a late Saturday video address.

Attempting to disrupt the Ukrainian redeployment, Russia has said that its forces carried out a series of airstrikes on Ukrainian railway hubs. Some 52 people died in Friday’s Russian missile attack on the railway station in the Donbas city of Kramatorsk just as it was packed with civilians trying to board evacuation trains toward the relative safety of western Ukraine, according to Ukrainian authorities. Moscow denied it carried out that particular strike.

Civilian evacuations continued Saturday from Kramatorsk, in eastern Ukraine’s Donetsk region.

Photo: Andriy Andriyenko/Associated Press

Authorities over the weekend urged all civilians in Ukrainian-controlled parts of Donetsk and Luhansk regions—which comprise Donbas—and two districts of the Kharkiv region to use all available means to leave immediately. They organized additional trains and buses.

Russia launched several additional missile strikes on the Dnipropetrovsk region just west of Donbas on Sunday, destroying an infrastructure facility in the city of Dnipro and an industrial facility in the city of Pavlohrad, the regional administration said. Ukrainian forces overnight destroyed a Russian column that was moving toward Izyum, according to the governor of Kharkiv. The claim couldn't be independently verified.

Russia’s Defense Ministry said it targeted Dnipro with Kalibr missiles from a Black Sea fleet frigate and that its missiles also struck the southern Ukrainian region of Mykolayiv and a military airfield near Kharkiv, where it said it destroyed an S-300 antiaircraft missile system. Those claims couldn't be confirmed independently. Dnipro—one of Ukraine’s biggest cities—is the logistical hub for Ukrainian military operations in Donbas.

Russia’s initial attempt to seize Kyiv and other cities in northern Ukraine collapsed in late March, in part because nimble Ukrainian units attacked Russian tanks and armored vehicles using guerrilla tactics, striking at Russia’s long supply lines that ran through woods and villages whose residents relayed intelligence to Ukrainian forces. Light, portable missiles supplied by the U.S. and its European allies, such as the Javelin and the NLAW, played a big role in that success, as did Turkish-made Bayraktar TB2 armed drones.

Satellite imagery showed a large military convoy moving south on Friday through Velykyi Burluk, a northeastern Ukrainian town near the Russian border.

Photo: -/Agence France-Presse/Getty Images

A destroyed apartment building in Borodyanka, a Kyiv-area town hit hard by Russian forces.

Photo: Alexey Furman/Getty Images

Many Russian battalion tactical groups that withdrew from northern Ukraine were battered so badly that they won’t be able to redeploy to the Donbas front anytime soon, Ukrainian and Western officials said. “We’ve seen indications on some units that are, literally, for all intents and purposes, eradicated,” a senior Pentagon official said. Russia, he said, is trying to mobilize some 60,000 reservists to fill the gap in manpower.

The tactical situation is more advantageous for Russia on the Donbas front. Russian supply lines are shorter, and the more concentrated area of operations allows Russia to more effectively use air support, Ukrainian and Western military officials said.

This different type of warfare, with large formations facing each other instead of small-unit strikes, is a major reason why Kyiv says it urgently needs heavy weapons, such as artillery, tanks and antiaircraft batteries that most Western allies have been reluctant to supply so far.

“The battle for Donbas will remind you of the Second World War, with its large operations and maneuvers, the involvement of thousands of tanks, armored vehicles, planes and artillery. And this will not be a local operation, based on what we see in Russia’s preparations,” Ukraine’s Foreign Minister Dmytro Kuleba said after meeting North Atlantic Treaty Organization ministers this past week. “Either you help us now—and I’m speaking days, not weeks—or your help will come too late and many people will die.”

While Ukraine initially sought Soviet-designed heavy weapons systems that its troops are trained to use, the limited supply of this equipment and ammunition, combined with the prospect of a lengthy conflict, mean that Kyiv is now requesting purchases of NATO-standard heavy weapons, Ukraine’s Defense Minister

Oleksii Reznikov said.

“The Soviet-made weapons that we have obtained can only strengthen Ukraine for a short time,” he said in a speech posted by the Ministry of Defense.

Ukraine managed to win the first round of the war because of close-contact infantry engagements, he said, but now Russia has changed its tactics and is relying more on long-range artillery, aviation and missile strikes—weapons that Ukraine has limited ability to counter.

“The war is entering the phase of competition for resources, which are almost unlimited in Russia in comparison to Ukraine,” Mr. Reznikov said. “To win in this war, we need a different kind of assistance from what we received before.”

Western leaders are stepping up support. British Prime Minister Boris Johnson, who Saturday took a walk around Kyiv with Mr. Zelensky, announced new military and other assistance that includes 120 armored vehicles and new antiship missile systems. That is on top of Friday’s package of Starstreak short-range antiaircraft missiles, 800 more antitank missiles and high-tech munitions that loiter above targets for precision strikes.

“This war will be won on the battlefield,” European Union foreign policy chief Josep Borrell tweeted on Saturday. He traveled to Kyiv on Friday, along with the head of the European Commission, which afterward said the EU is proposing 500 million euros, equivalent to $544 million, in new support for Ukraine’s military, on top of 1 billion euros previously allocated for weapons

A resident inspects the remains of her home after shelling on the outskirts of Chernihiv.

Photo: sergey dolzhenko/Shutterstock

The U.S. announced $400 million in additional military aid to Ukraine in April, out of a total of $1.4 billion since the war started Feb. 24. Washington is supplying Javelins, Stinger missiles, hundreds of Switchblade loitering drones, and counter-artillery radars—but, so far, no heavy weapons requested by Kyiv.

So far, only the Czech Republic has supplied Ukraine with tanks, sending Soviet-designed T72Ms, while Slovakia shipped to Ukraine its S-300 air-defense system after the deployment of Patriot batteries to replace it.

Before Russia’s full-scale invasion began in February, Ukraine controlled about two-thirds of the Donetsk and Luhansk regions that comprise Donbas. The rest were governed by the so-called Donetsk and Luhansk people’s republics, statelets created following Russia’s intervention in 2014, when Moscow also annexed Crimea from Ukraine.

Russian President Vladimir Putin in February recognized these statelets in their claimed borders that cover the entirety of the Donetsk and Luhansk regions, including the cities of Kramatorsk, Slovyansk and Severodonetsk that remain under Ukrainian control.

Ukrainian officials said Moscow’s aims likely go far beyond seizing these areas, and that Mr. Putin seeks to destroy the best Ukrainian units in the battle of Donbas to then try again to seize the rest of the country, including Kyiv.

Unlike in the northern regions of Kyiv, Chernihiv and Sumy, where support for Ukraine’s independence from Russia has long run high, Moscow can count on at least some cooperation from locals in Donbas and nearby areas of eastern and southern Ukraine, where some part of the population is sympathetic to Russia. In the Luhansk region, four city mayors have already switched sides and begun collaborating with the Russian military, including the mayor of the front-line city of Rubizhne, according to the regional administration.

A man in the Kyiv-area village of Andriivka looks at tanks shells left by retreating Russian forces.

Photo: Celestino Arce/Zuma Press

In besieged Mariupol, where fierce Ukrainian resistance continues in parts of the city, tying up a large Russian force, a member of the city council from a pro-Russian political party has assumed mayor’s duties under the auspices of Russian occupation forces, according to his appearances on Russian TV. Ukraine said it has begun treason proceedings against him.

At least 5,000 people, and possibly many more, have died in Mariupol under Russian shelling and bombing, according to the city’s elected mayor. Russian state media usually blame the destruction there, and elsewhere in Ukraine, on Ukrainian “Nazis” who allegedly shell their own cities to incriminate Moscow.

In an unusually frank admission, Aleksandr Sladkov, Russian state television’s military commentator, wrote in a social-media post that Mariupol is being leveled to present an example to the rest of Ukraine.

“Let them see in Kyiv and Lviv, Cherkasy and Poltava, Ternopil and Chernivtsi—if the city doesn’t surrender, it gets annihilated,” he wrote. “The cities of central and western Ukraine will also be destroyed if they decide to resist Russian troops.”

—Max Colchester, James Marson and Gordon Lubold contributed to this article.

Write to Yaroslav Trofimov at yaroslav.trofimov@wsj.com