Moving projects from one organization to another

I am in need of help.
We have created a design system (several Figma files) for a client, that our team owns. This system is then connected with many other product files within our Figma team (in other projects). We have self hosted the files within our team, due to the client not having Figma or having the need of it.
Some time has passed and our client now has need for Figma, for other project not related to ours.
Our job is now to migrate the design system and the product files to their Figma setup. We have other client so we don’t want to migrate the entire team, only the projects that are related to them. There are a lot of connections between the files, and we don’t want to reconnect them in the new Figma setup.

Anyone that has done a migration like this, and have some tips on a workflow?

Have the same problem in my workflow. I have three files in my own organization/team and each of them contains some components linked with other files. Now I need to move all projects to the client’s organization/team and I’m worried about all connections and links to the prototypes.

I contacted the Figma support and together with the admin of the other Figma organization, we did a migration of the files. It worked!
A thing to note is that we had several mail conversations before we did the migration. Just to make sure everybody was clear on what outcome we wanted to achieve.