There is no REST API for moving issues yet.
It's not something you should be automating or scripting for really, it's not something you should do regularly. If you are regularly moving issues between projects, you've probably got a broken process.
NOTE: bulk-move of issues may be something that is not done frequently, but if you need to do complex moves of many issues it can prove problematic to do so using the web UI even if it only happens occasionally. Take for example this use case: you import thousands of issues from another issue tracking tool into Jira and, for simplicity, you dump all the new issues into a single project. However, afterwards suppose you want to split up that data between dozens of new projects in Jira. Trying to create JQL queries for each one of the secondary projects, and manually migrating the issues one project at a time, would be very time consuming and error prone. It would be very handy to have support for this added to the REST API.
This is an old question but I have this problem and it's to Move issues from one issue type to another within a project as part of cleaning up issue types. For example someone created "Sub Task - QA" in the past but we want everyone to just use "Sub-Task". In the UI, changing issue type within a project is a Move and has the same problems. The other way is to update the issue type scheme - removing the old type - which forces a bulk Move, but that has scalability issues and does not work for more than 1000 issues.
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