Could someone explain how a Continuous Overseas Tour works? Korea to Germany?

You will need to go through your S1 to do a 4187 requesting a Consecutive Overseas Tour (COT).

It was called consecutive overseas your in my day and then it worked like any other PCS assignment except you were going to (possibly) another country. I went from Thailand to Vietnam and had orders to Japan after that. So if your family is with you in KR, they will move with you to DE. If you are on an isolated/remote tour, The move will like going stateside except you wife will need to deal with passports, visas, possibly, contacting movers, etc. But that was then, Now could be the same...or not. It's gonna be fun, enjoy it.

That’s a good question. Either check with Mil Pers or your Top. They know everything! Wish you the best!