I'm looking for way to catch "issue moved" event in Jira Cloud and send him by webhook.
It's possible? From my research results that no, but maybe someone have a workaround for this?
Yep. There are some apps which can provide listeners to catch events and fire off webhooks, but you can do the basics with automation too.
If you go to Admin -> Automation (or project settings -> Automation), and add a new rule, you should find a trigger called "when issue moved", and an action of "send web request" with parameters for the call, content and type. Those two combined should do what you need.
Yes, and... to what Nic suggests regarding use of the issue moved trigger for Automation for Jira...
There are some open defects with that trigger, so take a look in case any of these impact your use case implementation:
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