Has anyone faced a similar issue where Zeebe stops exporting data, and the data size continues to increase, particularly when the backup API is triggered?
During this time, Zeebe becomes unresponsive, and it appears that the backup process does not start.
Additionally, while retrying the backup, CPU consumption increases significantly.
If we resume exporting data to Elasticsearch, will the backup process start automatically, or does this require further investigation into other components?
This sounds like a known issue where the exporter gets stuck during backup operations, causing Zeebe to become unresponsive and preventing the backup process from completing properly. I found the following relevant resources:
Does this help? If not, can anyone from the community jump in? 
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No, actually zeebe not able to export data to elasticsearch because elasticsearch shards limits exceed to couldn’t able to flush data
Then I observ zeebe components backup stopped as backup api not responding