You can use a csv import for the data and then manually copy the other stuff. Or you can look into the JSON option:
If you are growing and scaling your usage, it might be a good time to look into using server. Cloud has a 25g limit on storage and about a 1 gb cap on the JVM.
To jump onto Ryan's answer, it is due to the lack of support for single project import into Cloud that would require you to import the issues individually as he suggested.
For more information or for alternative solutions, you can have a look at How to migrate projects from one JIRA Cloud application to another.
Finally, we do have a feature request below, that you can vote on and watch for any updates or changes:
Let us know if you have any questions or trouble with anything we've gone over so far.
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