It depends if the site you are moving to is a new and empty site or already has data. If your target site is empty, the best way to transfer is via backup and restore. See more information
If the target site already has data in it, you are essentially looking at merging sites. You can find more information about available methods in this support article.
It may be good to know that a cloud to cloud migration assistant is on the roadmap for Q2 2021 and can be expected soon. As you can see in JRACLOUD-72570 an early access is already available. You can sign up from a link in the ticket.
Hope this helps!
Hi @ondrej.horacek I have quite fresh experience with Atlassian support. They are able to move complete instance to new URL if you need this. It is described here https://confluence.atlassian.com/cloudkb/changing-the-cloud-instance-url-address-or-migrating-to-a-new-cloud-instance-691011835.html but the problem can be, that your instance is on Free plan.
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