Ukraine has increased its frequency of drone attacks on Russia in recent weeks, a tactic U.S. officials say is intended to demonstrate to the Ukrainian public that Kyiv can still strike back, especially as the counteroffensive against entrenched Russian troops moves slowly.

Over the past week, Ukrainian drones near Moscow forced the Kremlin to temporarily shut down airports serving the capital. And on Friday, the Russian Ministry of Defense said Ukraine had launched 42 drones at the occupied Crimean Peninsula and fired a missile that was intercepted not far from Moscow, in what could be one of the biggest known aerial assaults on Russian-held territory since the war began.

Throughout the summer, the intensifying strikes — many of which have been carried out with Ukrainian-made drones — have hit a building in central Moscow, an international airport and a supersonic bomber stationed south of St. Petersburg.

Although the attacks destroyed the bomber, they have done little significant damage to Russia’s overall military might, U.S. officials have said. No Russians have been killed in the strikes on Moscow, most of which occurred early in the day, reducing damage and disruption. The timing may be for operational security or to avoid Russia’s air defenses, but it has also helped ensure that the attacks did not prompt escalatory attacks by Russia.

Andriy Yusov, a spokesman for Ukraine’s military intelligence service, known as the G.U.R., did not directly claim responsibility for the attacks, but he said strikes on Moscow would continue.

“Russian elites and ordinary Russians now understand that war is not somewhere far away on the territory of Ukraine, which they hate,” Mr. Yusov said in an interview last month, as the drone campaign began to intensify. “War is also in Moscow, it’s already on their territory.”

But U.S. officials say there is a more important audience. If there is a strategic target, it is to bolster the morale of Ukraine’s population and troops, according to the officials, who spoke on the condition of anonymity to discuss sensitive information.

Ukraine began a counteroffensive in June against Russian forces occupying its south and east. But unlike the push last summer, during which Ukrainian forces quickly retook land outside Kharkiv, they have found it harder to break through Russia’s fortified defenses and incurred heavy losses in equipment and troops.

Without appreciable progress to demonstrate on the battlefield, Ukraine has intensified its drone attacks in Crimea, in cross-border strikes and deep into Russia.

Ukrainian officials said they had some hope that stepped-up attacks could force Moscow to reconsider its far more extensive and damaging missile and drone strikes on Ukraine.

But so far, the campaign appears to have strengthened support for the war in some parts of Russia, though it may have also helped undermined faith in the government, at least in elite circles, said Tatiana Stanovaya, a scholar at the Carnegie Endowment for International Peace.

“The Ukrainians do what they can in their current circumstances,” said Ms. Stanovaya, who based her assessment on reporting and conversations with people in Russia. “I don’t think they are trying to have a strategic effect.”

In Russian border regions, the drone strikes appear to have bolstered anti-Ukraine sentiment and may help strengthen President Vladimir V. Putin’s standing, Ms. Stanovaya said. “It fuels fears among Russians that they are vulnerable, they can be attacked, so they have to support Putin,” she said.

But in a recent article in Foreign Affairs, Ms. Stanovaya argued there were signs — at least among wealthy elites — of doubts about Mr. Putin’s strength after the rebellion by the mercenary Wagner group. The intensifying Ukrainian drone strikes may be contributing to those doubts.

U.S. intelligence agencies are unsure which operatives in the Ukrainian military are carrying out the attacks. Though he has not mentioned the strikes specifically, President Volodymyr Zelensky of Ukraine has said the war is returning to Russian territory.

Ukraine has used a mix of short- and long-range drones to conduct the strikes, according to U.S. officials. A New York Times investigation last month found that three types of long-range drones made in Ukraine had been used in the attacks on Russia, including the Moscow region. Those drones were launched from Ukraine, U.S. officials said. The short-range ones are likely to be operated by pro-Ukrainian sympathizers or operatives who slipped across the border. Some of the drones were built in Ukraine, the officials said, while others might have been assembled in the field. In either case, the United States believes the attacks were most likely ordered and loosely directed by elements in the Ukrainian government.

U.S. officials said the cross-border strikes were not carried out using American equipment.

The first major strike in Moscow — in early May, when drones exploded over the Kremlin — was possibly halted by Russian air defenses. In July and August, several more strikes have occurred in and around Moscow.

Despite the limited effect of the attacks, U.S. officials who spoke on the condition of anonymity said they expected Ukraine would continue to strike in Russia because it sent a strong message: Kyiv was not simply defending its territory but also had some ability to take the fight to Russia.

Earlier in the war, U.S. officials had warned of the possibility that Ukrainian drone strikes on Moscow could prove escalatory, giving Mr. Putin an excuse to intensify Russian attacks on civilian targets. But U.S. officials conceded that attempted Ukrainian strikes had so far been calibrated, and they had not provoked any drastic escalation by Moscow.

Ukrainian officials said the drone strikes posed little risk of escalation because Russia could not further intensify its fight given that it was already firing as many missiles and drones at Ukraine as it could.

Analysts said the drone attacks had propelled the Russian propaganda machine into action. State-run media in Russia has sought to blunt the impact of Ukraine’s drones by emphasizing the military’s ability to shoot them down, an attempt to send a message that its defenses were strong and the Ukrainian strikes ineffective.

“The narratives that are coming out of Russian media, reflective of how the Kremlin is trying to narrate it, are referring to drones shot down, not drone attacks,” said Jonathan S. Teubner, the chief executive of FilterLabs, a company that tracks public sentiment in Russia. “They have been communicating, ‘We have shot down another Ukrainian-Western-American drone, look what we’ve done.’”

Mr. Teubner said his company’s studies of Russian sentiment, which use computer models to analyze comments on Telegram, social media, local messaging groups and other sources, show the population is largely fed up and disgusted with the military leadership in Moscow, especially after Yevgeny V. Prigozhin’s attempted rebellion in June.

While the abortive mutiny has helped accentuate public skepticism of a draft in Russia, it has not yet turned ordinary Russians against the war, Mr. Teubner said. But the drone strikes, combined with the aftermath of the rebellion, may have increased the view among Moscow’s elites that the Russian government could be more frail than it seems.

U.S. intelligence officials had predicted for weeks that Mr. Putin would not allow Mr. Prigozhin’s mutiny to go unanswered, in part because to do so could undermine the public’s view of the Russian government. U.S. and Western officials said they believed Mr. Putin had ordered the destruction of Mr. Prigozhin’s plane. The crash appears to have killed the mercenary leader.

For now, U.S. officials say it is important not to overstate the effect the drone strikes are having on Russian public opinion.

Russia has executed a similar tactic, but with a far larger barrage of attacks that have far more lethal consequences. Since last fall, Russia has used Iranian drones and its own cruise missiles to strike apartment buildings, power stations and other civilian infrastructure in Ukraine.

Those attacks, which have killed scores, are part of a campaign by Russian forces who hope the strikes will sap the will of Ukrainians to resist the invasion, U.S. officials said.

Yet despite widespread damage, the Russian strikes have had a limited impact at best on Ukraine’s military might and its will to fight. Even the strikes on the electric grid have had only transitory effects. In the largest cities Ukraine, power was restored after Russian attacks on the electric grid in six hours on average, an analysis by Microsoft showed. In other areas, power was restored in 3.3 days.

Military commands often justify strategic bombing campaigns on civilian populations as a way to break the enemy’s will to fight. But such attacks backfire more often than they succeed, U.S. military officials said. Germany’s attacks on London during World War II, for example, hardened Britain’s resolve.

“There is very mixed data over whether bombing a civilian population changes their attitude in the way you want it to be changed,” Mr. Teubner said. “In the London Blitz, the Brits didn’t turn to Churchill and say, ‘We don’t want bombs coming down.’ It had the opposite effect of what the Nazis wanted.”

U.S. officials said some Ukrainian officials understood that just as Russia’s attacks on Kyiv would only strengthen Ukraine’s resolve, Ukraine’s attacks on Moscow were unlikely to turn Russians against the war.

Instead, the symbolic strikes are meant to give heart to Ukrainians who may be anxious about the slow progress of the counteroffensive, buying time for Mr. Zelensky and his forces in the coming weeks.

Christiaan Triebert and Andrew E. Kramer contributed reporting from Kyiv, Ukraine, and Anton Troianovski from London.