Yes, the Kibana dashboards are being saved in Elasticsearch under kibana-int index (by default, you can override that in the config.js file). If you want to move your Kibana dashboards to another ES cluster you have two options:
- Export manually the dashboards. Click on Save -> Advanced -> Export Schema. You have to save the file and then in the new Kibana you have to import click over Load -> Advanced -> Choose File and choosing the file that have selected. This is a pain, because you have to do this operation per dashboard that you want to migrate.
- You can use an utility to migrate an ES index from one ES cluster to another. There are some utilities already that can perform this operation. Searching in SO, I found this answer that suggest you to use the Elasticsearch.pm library (Perl :S) to do this. Probably there are more utilities like this but I do not think that doing a script that migrate an index to another cluster is such a difficult task.
EDIT: For the second option, you can use the python elasticsearch library and its helper reindex, if you feel more confortable with Python: https://elasticsearch-py.readthedocs.org/en/latest/helpers.html#elasticsearch.helpers.reindex