Tillerson says diplomacy with North Korea has‘failed’; Pyongyang warns of war

But to address your question, in the event of war, Non-combatant non-Korean native personnel would be evac'd relatively quickly. I wouldn't foresee a travel ban.

North Korea doesn't let its people out now. In the event of a war they definitely won't let anyone out. Those people think the world outside of North Korean borders is some kind of terrible place anyway so I doubt they'd make an effort to get out.

A good question to ask, but any war with Korea will be short lived. The North's "tech" is mainly boots on the ground. They could overrun a Southern strong point or two, but the allies have had years to dig in, register fire, etc. Southern artillery would funnel them into kill zones if the USAF didn't get to them first. I'm sure we already have subs and other tech assets close by. One of the big war planners in the 80's (I believe Norman Polmar) surmised that once the border opened and the Northern troops ran into their first full scale bomb-beating, the Northern civilians would flood the roads enmasse like the European refugee's during the 1940's Blitzkrieg.

Should that happen, Northern efforts to reinforce destroyed/cut-off units would fail. Too many starving Northerners would bring any Northern military effort to a halt if not a slow crawl. And we are too successful in aerial killing (Desert Storm, etc) for you to be on the road doing anything other than attacking us. I give them a week at best. PacFleet SSGN's would Tomahawk their missile facilities to death. Navy DDG's have enough missiles to take out any nuke they launched. And Army Apaches and Marine Sea Cobras would bring the rest of the thing to a "made-for-tv" close.