The Kremlin has dispatched still more forces to shore up its faltering war effort, but the units are headed not to Ukraine but to Russia’s borders with other countries, where on Tuesday they were confronting young Russian men trying to join an exodus out of the country.

As the avenues for Russians to escape a draft order issued last week narrowed, the Federal Security Service sent armoured vehicles to the frontiers, where some men waiting to flee were being served military call-up papers, the state news media reported.

The rush to the borders began within hours of President Vladimir V. Putin’s announcement last week of a military call-up affecting hundreds of thousands of Russians, and the flow has got only bigger since then.

On Tuesday, at the borders of Georgia, Kazakhstan and even Mongolia, their numbers continued to swell.

In Kazakhstan, responding to calls that the 4,600-mile border with Russia is closed, President Kassym-Jomart Tokayev urged “humanity, patience and organisation”, saying the Russians had been “forced to leave because of the current hopeless situation”.

With cars lined up for miles at its border and waits of more than 48 hours, Georgia said it would allow visitors to enter on foot. The number almost doubled over the past week, to about 10,000 a day.