The fact that Dutch people staying behind in Joseon after Hamel and his mates escaped from Joseon were sent back home via Japan is a very good example in understanding international relations in those days.

It was because this negotiation was not made in direct way between Joseon and Netherlands, but in indirect way via Japan.

Dutch people shipwrecked in Joseon could return home with the existing foreign relations between Japan and Netherlands, not between Joseon and Netherlands.
Also, the relation between Joseon and Japan was a precondition of sending Dutch people back home.

In other words, when castaways arrived in a country where they have no foreign relations, they could return home via the third country such as Japan which they have foreign relations with.

Such structure of sending castaways back home and international relations can be displayed in graph form as follows.

Seven(7) Dutch men staying behind in Joseon could go to Japan because there was state authority of Joseon and Japan. In addition, international relations between Joseon and Japan and between Japan and Netherlands facilitated their return back to Netherlands.

The repatriation system was not established between Japan and Netherlands until 1643. Since Dutch ship, the Breathkense was shipwrecked in Yamada(山田) in 1643, the Japanese government made the standards of dealing with the shipwreck of Dutch ships.

In 1627 before the Breathkense sudden arrival, Weltevree (Park Yeon) arrived in Joseon and the Joseon government wanted him to handed over to Japan but Japan rejected it.

At that time, although Japan had relations with Netherlands, Japan had to reject because "the system of sending castaways back home" was not yet established between Joseon and Japan and between Japan and Netherlands.

But with regard to Hamel and his mates in 1653, Japan requested Joseon to send them back to Japan.

Although they were shipwrecked in Joseon, Japan could request their repatriation having confirmed their stay in Joseon.