ZAPORIZHZHIA (UKRAINE) : Civilians trapped for weeks in the rubble of Mariupol and other towns occupied by Russian forces are trickling into Ukrainian-controlled territory after sometimes long and traumatic journeys, while Russia continues its offensive in Ukraine’s east and targets the strategic port city of Odessa.

       A group of civilians from Mariupol, including around 100 who were trapped in the besieged Azovstal steel plant, were due to arrive at a processing center in Zaporizhzhia, a Ukrainian-held city in the country’s southeast, on Tuesday morning after a two-day journey through territory held by Russian forces.

       Many of the evacuees are traveling in buses overseen by the United Nations and the Red Cross, which are trying to broker safe passage out of Mariupol for remaining civilians.

       The convoy from Mariupol had been stopped en route to Zaporizhzhia on Monday evening at a Russian filtration camp in the town of Manhush, a U.N. official said. Civilians who have made it through Manhush to Zaporizhzhia in recent weeks said Russian forces at such camps check evacuees’ phones for evidence of any connection to the Ukrainian military or of efforts to pass information to Ukrainian forces. Evacuees report that many men are being detained at the filtration camps.

       The evacuation effort from one of Ukraine’s worst-affected cities came amid continued Russian strikes elsewhere in the country. Russia on Monday repeatedly fired cruise missiles at Odessa. An afternoon strike, which hit a Russian Orthodox church, killed a number of civilians, including a 13-year-old child, according to the local authorities.

       Earlier in the day, a Russian strike targeted a strategic bridge near Odessa, the third such attack on the crossing in recent days. The bridge provides the only internal connection to the western part of the Odessa region that faces Snake Island, a position held by Russian forces since February. The area is also reachable over land via Moldova.

       On Tuesday, Russia’s Defense Ministry said it had launched cruise missiles at a military airfield in Odessa that was holding foreign weapons. It also said it had destroyed an S-300 air-defense system in the nearby Mykolayiv region and an ammunition depot near the northern city of Kharkiv. The claims couldn’t be independently verified.

       Russia’s offensive to seize swaths of Ukraine’s east has largely stalled amid fierce Ukrainian resistance and heavy Russian losses, military experts say. The General Staff of Ukraine’s armed forces on Tuesday said Russian forces had suffered 23,200 combat deaths since the start of the invasion on Feb. 24. Russia on March 25 said 1,351 service members had been killed. It hasn’t released a death toll since.

       Western officials say Russian President Vladimir Putin is under pressure to announce some kind of battlefield victory on May 9, when Russia celebrates Victory Day to commemorate Soviet forces’ victory over Nazi Germany in World War II.

       But apparent strikes against targets on Russian territory in recent weeks signal an escalation in the war, with explosions ripping through Russian ammunition depots, fuel facilities and railway bridges in the border regions of Belgorod, Bryansk and Kursk that serve as logistics bases for Russian forces attacking Ukraine. There has also been a series of unexplained fires at military installations deep inside Russia, raising questions among local residents who have posted videos of the conflagrations to social media.

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       The column of buses from Mariupol that was expected in Zaporizhzhia has been joined on its route northwest by residents leaving other Russian-occupied towns, including Tokmak and Vasylivka. Many other civilians trying to escape have joined the column in private cars or other buses arranged by local authorities, said a Ukrainian police officer who is helping organize their arrival and processing in Zaporizhzhia.

       “Anyone who wants to and is able to evacuate can now step outside and await the column," Zaporizhzhia regional governor Oleksandr Starukh wrote on his Telegram channel on Tuesday morning, addressing residents of Tokmak and Vasylivka. “We’re waiting for you!"

       In Zaporizhzhia, Mariupol residents who have evacuated their city, largely taken over by Russian forces after weeks of intense bombardment, have recounted harrowing efforts to survive amid dwindling food and water supplies, as well as perilous, multi-day trips to the relative safety of Zaporizhzhia. The trips require passing multiple Russian checkpoints, where male evacuees are instructed to strip off all clothing except for their underwear and pants, which Russian soldiers roll up to the knee checking for tattoos suggestive of pro-Ukrainian sentiment.

       “They told us to get out of our cars, lined the men up and shouted at us to take off our clothes," said Alexey Miroshnichenko, a construction worker who left Mariupol with his wife, Sevel, and 3-year-old son, Lev, and traveled with six other people in a minivan driven by a local volunteer. “They rifled through our bags at every checkpoint we passed."

       Each day in Zaporizhzhia, a Ukrainian city on the edge of Russian-occupied territory that serves as a waypoint for people fleeing the violence, families who managed to escape occupied territory stop off at the large parking lot of a shopping mall on the city’s outskirts. There they are processed and registered as refugees. Police officers check their documents before ushering them to a tent where volunteers offer free food and advice on how to reach other Ukrainian cities and find accommodation there. One half of the tent is taken up by stalls handing out items of second-hand clothing and children’s toys.

       The majority of the cars pulling up at the parking lot are filled with families, often with pets in tow. Often there are minivans driven by volunteers, like Kseniya Safonova, a 24-year-old Mariupol resident who spent the first weeks of the war bringing aid to Ukrainian soldiers fighting off the Russian advance and later helped fleeing families to find transport connections out of the city.

       “I felt that we have to help those who defend us and ensure that we can live in a free country," Ms. Safonova said. On April 26, she left Mariupol, pledging to return to the city when it has been liberated from Russian troops. It took her almost a week to reach Zaporizhzhia.

       The Mariupol evacuees described a city subsisting on scant Russian aid deliveries after weeks of bombardment that left its residents cooking scraps of food on makeshift stoves in the courtyards of accommodation blocks gutted by tank and artillery fire.

       Korneliusz Wieteska, a Polish businessman who hired a bus and drove from Opole in Poland to help transport Mariupol evacuees, said many Ukrainians coming from Russian-occupied territory are so traumatized by their experiences that they decline free accommodation arranged in hotels en route and ask him to drive through the night to Poland because they are desperate to leave Ukraine.

       Valentina Portyanchenko, 75, arrived in Zaporizhzhia on Monday after being persuaded by her daughter to leave Mariupol despite her desire to stay. Ms. Portyanchenko said that when she was four years old her mother, who survived a Nazi concentration camp, brought the family to live in Mariupol several years after the end of World War II. Ms. Portyanchenko said the hardships she endured as a child paled in comparison with the desperate shortages of food, water and power caused by the city’s recent destruction by Russian forces.

       “Everything is destroyed," she said. She began weeping when she said she wasn’t even able to pack clothes in her rush to leave, and those she had traveled in—a tattered shirt and cardigan—had been donated by volunteers like Ms. Safonova, who helped spirit her out.

       Mariupol is now under the control of Russian forces, evacuees said, with a strict system of permits in place for anyone wanting to move around the city and reliable cellphone connection restored only for owners of a Feniks SIM card. The network operator is based in the pro-Russian breakaway states of eastern Ukraine and the only carrier now allowed to register clients in Mariupol.

       The Azovstal steel plant is the final holdout in Mariupol for Ukrainian soldiers who have spent weeks defending the city. According to accounts from those who have fled, the plant is being hit daily by Russian airstrikes and artillery, causing reverberations throughout the compound that register even several levels below ground, where most of the roughly 900 civilians remaining at the plant are taking refuge.

       —Evan Gershkovich contributed to this article.

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