KYIV — As Ukraine prepares to ramp up military conscription to defend against Russia’s invasion — after a key deadline to register with recruitment offices passed this week — some men are dodging the draft preemptively by paying thousands of dollars for help to illegally leave the country.

       Smugglers who assist with border crossings often charge more than $5,000, according to Ukrainian officials and men who have paid for the service. One man was driven by bus with others to a forest. Guides wearing masks then led them on foot to breach a fence at the Hungarian border.