KYIV, Ukraine—As soon as the military draft officers limped into an outdoor shopping center here, young men began moving away from them. The first man they tried to stop held a phone to his ear and kept walking past.

“They’re afraid to go fight,” one of the recruiters, a 36-year-old who goes by the call sign Fantomas, said.

Ukraine is in the midst of a massive mobilization drive, with the goal of adding hundreds of thousands of new troops to the country’s forces. New laws have lowered the age of conscription from 27 to 25, and imposed harsh new penalties on men who fail to update their draft registration. But unlike in the early days of the war—when so many Ukrainians volunteered after Russia’s invasion that the army had to turn some away—many men of fighting age are trying to avoid the military by any possible means.

The Ukrainian military recruiter, known by the call sign Fantomas, began working as a draft officer after injuries forced him off the battlefield. Photo: Joseph Sywenkyj for WSJ

Applications for postgraduate education programs, which traditionally came with an exemption from the draft, reached record highs this year. Smugglers charge up to $15,000 to get men out of the country illegally. Some men have effectively gone into hiding.

Their efforts to avoid fighting highlight an uncomfortable reality for Ukraine in the third year of full-scale war: Most men who wanted to join the armed forces have already done so, and enlisting more men is likely to become increasingly difficult the longer the war goes on. Already, troops at the front in eastern Ukraine complain that their battalions are short of men and in desperate need of reserves.

Draft dodgers list a host of reasons for trying to stay out of the military. Many have heard stories of men who were sent to the front with minimal training, and they fear that commanders will use them as cannon fodder. In addition, once they are drafted, the only currently available ways to leave the military are injury, death or the end of the war.

“It’s a big deterrent for me,” Oleksandr, a 25-year-old in Kyiv, said of indefinite enlistment. “If you die, what happens to your family?”

In June, Oleksandr quit his job at a bakery, worried that he would eventually be caught by draft officers if he kept commuting to work. Now, he is living off savings and hardly leaving his apartment.

“It’s very dangerous,” Oleksandr said of going outside. “I go to the nearest store for groceries. I don’t go out for walks.” (Like others who are trying to avoid the draft, Oleksandr requested that his last name not be used.)

A recruitment poster for a Ukrainian army battalion hangs in Kyiv where the police, working with military recruiters, check the documents of young men. Justyna Mielnikiewicz for WSJ

Dmytro Lazutkin, a spokesman for the Defense Ministry, said the mobilization drive is going well: 4.6 million men have updated their data since the new law was passed, and more than 3,000 people have voluntarily applied to join the armed forces over the last three months, a sharp increase. He declined to comment on how many men had been added to the armed forces overall.

Lazutkin told Ukrainian media that stories of people being sent directly to a trench are untrue.

“First, there’s basic military training,” often followed by more specialized training, he said. “The percentage of those who participate in assault operations, out of the total number of servicemen, is actually not that high.”

Still, there are indications that huge numbers of people are trying to avoid the draft. More than 90,000 men of conscription age applied this year to enroll in postgraduate programs, which had previously come with a draft exemption. That exemption has now been largely eliminated, the Ministry of Education said, “so that postgraduate study does not turn into a corrupt tool to avoid mobilization.”

Avoiding conscription is growing tougher by the day.

Draft officers have fanned out across the country handing out summonses. After receiving a summons, men must appear at a recruitment office within days, or a warrant goes out for their arrest.

Military recruiter Fantomas checks draft documents as he walks around the streets of Kyiv. Joseph Sywenkyj for WSJ

Nazar, 39, was living in Uzhhorod, in western Ukraine, when the new mobilization law was approved earlier this year. Soon after, he said, draft officers began blanketing the city. Several of his friends were arrested after failing to respond to summonses.

“They were everywhere—especially at the entrances to the city and the train station,” Nazar said.

Nazar said he didn’t want to fight, adding that his cousin and several friends had already been killed in the war. So he quit his job at a shop in Uzhhorod, moved to Kyiv and started working on a construction crew.

For much of the war, Kyiv was seen as a place of relative safety from the draft, with far fewer officers doling out summonses. But that has changed over the past few months.

In May, Nazar and three co-workers were waiting for a train after finishing work when an officer approached them on the platform and handed them all summonses. He quit several weeks later and has since holed up at his apartment, going outside as little as possible and trying to find a job that will give him an exemption from the draft. He realizes that a warrant is likely out for his arrest.

“I love my country. I have no problem with the authorities. I understand the need for mobilization,” he said, but added, “I hope that the war will be over soon and I won’t have to go. I’m not ready.”

Many Ukrainian soldiers—who, in many cases, have fought for more than two years with little rest—have grown resentful toward compatriots who are hiding or fleeing the country.

“I don’t understand people who don’t want to protect their mothers, their wives, their kids,” Fantomas, the draft officer, said. At the start of the full-scale invasion, he said, he quit his finance job and stood in line for two weeks to join the military.

Young men are notably few on the subway in Kyiv. Recruitment posters for Ukraine's army appear in the streets. Joseph Sywenkyj for WSJ

On a recent Thursday afternoon, The Wall Street Journal accompanied Fantomas and his partner on their rounds in Kyiv. A police officer trailed behind them—when recruiters find a man who already failed to respond to a summons, only the police can arrest him and take him to a draft office.

Both men began working as draft officers after injuries forced them off the battlefield. They didn’t chase after the men who ran away—Fantomas has a metal plate in his leg and walks with a distinct limp. A group on social media, with more than 200,000 members, tracks the movements of draft officers in Kyiv to help people avoid them.

The first man they successfully stopped was a heavyset 46-year-old, who said he had left his draft documents at home. Fantomas wrote him a summons and told him he had three days to appear at a draft office.

Fantomas took a photo of the man holding the summons to prove he hadn’t mistreated him. Videos online show draft officers forcibly hauling men into cars.

In a typical day, Fantomas said, he usually speaks to about 100 men and issues a dozen summonses; the police take several others, who have outstanding warrants, to draft offices.

The results are now reshaping life in much of Ukraine. Young men are notably few on the subway in Kyiv and the seaside resorts in Odesa. Some villages in western Ukraine have been effectively emptied of men. Businesses complain that it has grown more difficult to find workers in recent months, especially in traditionally male fields such as construction.

Yaroslav, 28, left his job at a graphic-design firm in May, when the company told him that all employees would have to prove that they had updated their draft registrations.

Now, he is living with his ex-girlfriend in Kyiv and working freelance design jobs from home. He no longer takes the subway or goes out jogging, and he has gained about 15 pounds in the past three months.

“My father condemns me for my anxiety of being mobilized,” he said. “But I’m simply afraid.”

People in Dnipro in southeastern Ukraine at a cemetery where Ukrainian soldiers are buried, and by the city's river. Justyna Mielnikiewicz for WSJ