My guess would be that you can download the backup from your free cloud and later upload it to the new environment (unsure about database versions on different clouds). A free plan can be upgraded to a paid plan, I expect that this would not destroy your data. It would still be in the public cloud though. If you want to migrate to a private cloud, maybe it is best to contact Mendix support, so they will be able to guide you.

Hi,

Please follow the instructions here as it explains the database steps aswell.

https://docs.mendix.com/developerportal/deploy/migrate-free-app-to-basic/