Is there a way to "migrate" a database in Pulumi, including data? I have tried the "Copy"

CreateMode

but after I've copied it and remove the "Copy" create mode, it thinks it's a state change and wants to delete that database and create new one.

You can using automation api https://github.com/pulumi/automation-api-examples/tree/main/dotnet/DatabaseMigration

cool, thanks, I'll take a look

While you can do a lot with the Automation API, having to implement this full flow is IMO a bit beyond what Pulumi is intended for. If you are on MySQL, I would use Pulumi to roll out Vitess.io and let the DB people use Vitess to migrate data between servers.