•Some have swum dangerous rivers to leave the country. Others have simply walked out under cover of darkness.
•After Russia's invasion, most men aged 18-60 were banned from leaving. But data obtained by the BBC reveals dozens have made it out daily.
Nearly 20,000 men have fled Ukraine since the beginning of the war to avoid being drafted, the BBC has discovered.
Some have swum dangerous rivers to leave the country. Others have simply walked out under cover of darkness.
Another 21,113 men attempted to flee but were caught by the Ukrainian authorities, Kyiv confirmed.
After Russia's invasion, most men aged 18-60 were banned from leaving. But data obtained by the BBC reveals dozens have made it out daily.
We have spoken to several men who have escaped in order to join family abroad, study, or simply make a living.
"What am I supposed to do [in Ukraine]?" one man, Yevgeny, said. "Not everyone is a warrior… you don't need to keep the whole country locked up. You can't lump everyone together like they did in the Soviet Union."
The BBC has established - by requesting data of illegal border crossings from neighbouring Romania, Moldova, Poland, Hungary and Slovakia - that 19,740 men illegally crossed into these countries between February 2022 and 31 August 2023.
While we do not know how those men escaped, we do know what methods were used by the other 21,113 who were caught trying. The majority - 14,313 - were attempting to walk or swim across the border, and the remaining 6,800 relied on fraudulently obtained official paperwork stating fake exemptions such as fabricated illnesses, the Ukrainian authorities said.
Those who are excluded from conscription include men with medical issues, those with caring responsibilities, and fathers to three or more children.
In August, Ukrainian President Volodymyr Zelensky called out the "corrupt decisions" made by the country's medical military commissions, which he said had resulted in a ten-fold increase in exemptions since February 2022. He announced that all regional officials in charge of military conscription had been removed, and more than 30 people faced criminal charges.
The president's parliamentary representative, Fedir Venislavskyi, acknowledged to the BBC that the problem was serious.
"The government realises that this phenomenon is not isolated and that it is widespread. But unfortunately, I would emphasise that corruption is very resilient," he said - adding that Ukraine was doing "everything possible to keep the number of corruption cases to a minimum".
Mr Venislavskyi said the number of men who had left or had tried to leave was having no impact on the war effort.
"I am convinced that the resilience and readiness of Ukrainians to defend their independence, sovereignty and freedom is 95-99%," Mr Venislavskyi told the BBC.
"Those who try to avoid mobilisation are about 1-5%. They are definitely not critical to the defence of Ukraine." He said there were no plans to radically increase the number of those eligible for mobilisation.
The 40,000-plus number of men who have fled, or tried to flee, could represent a significant proportion of the men Ukraine needs to replenish its army. In August, US officials estimated the Ukrainian military death toll to be up to 70,000 - although Kyiv won't give a figure.
The country also does not release official figures on the size of its army. But the new Defence Minister, Rustem Umerov, told the Yalta European Strategy forum in September that there are more than 800,000 in the Ukrainian armed forces.
Some of the escapes have been dramatic.
One video shows a man swimming across the Dniester River towards Moldova, with Moldovan border guards urging him across to safety. Another shows the potentially fatal consequences - bodies of men being pulled ashore, having drowned trying to cross the Tisa River between Ukraine and Romania.
But Yevgeny, a construction worker from Kyiv who we met in a Moldovan immigration centre, said he simply walked across that country's border - the most popular route out, our figures suggest. It is then relatively straightforward for escapees of the war to claim asylum.
Yevgeny had felt trapped in Ukraine he told us - younger men and those with military experience had been called up for conscription first.
It had been difficult for him in the meantime to find a well-paying job, "because everything is geared towards the war" and yet "electricity, fuel - everything's become more expensive".
After being processed by the Moldovan police, he applied for asylum - something that must be done within 24 hours of entering the country to avoid a criminal record.