I know for a fact that saved Kibana dashboards (ie, the JSON file of the dashboard) are saved in OR associated to a particular ElasticSearch
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:
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