Hello,
If your referring to Save search ( settings)? Then MongoDb is the place to be. Execute a mongodump. then SCP to new instance done.
If you referring to Data the Elasticsearch is the place to be, this would need to be. Execute a repo and move it to the new instance.
You may want to look here. Also there are a few demos in the forum on the “HowTo” /w ES and Mongo migration
Last, Migration of a widget, dashboard, etc… I would create a Content Pack. Download it and then Upload the Content Pack to the new server.