The Korean State Railway does an 8 day trip from Pyongyang to Moscow (and then back) via the Trans-Siberian route. Demand is kind of seasonal since the winter is rough, but the train usually leaves Pyongyang close to full of passengers headed to closer stops. Since it takes 8 days from end to end, it only runs roughly twice a month. You can take a passenger train or overnight train another 8 hours from Moscow to Minsk, multiple run daily.
Air Koryo also flies 1-2 times daily to Vladivostok and Beijing. You can get a connecting flight through there to Moscow and then to Minsk.
If you're asking how easily the average North Korean can get a passport and ticket to a destination outside the country, plane travel is often too expensive for most NK workers, but the train is heavily used.
Former Soviet Union states are typically the most desirable destinations for North Koreans. Kyrgyzstan has a large Korean population and offers unlimited visa-free stays for North Korean citizens. Most of the other CIS states grant eVisas for 30-90 day stays.