The field formats are part of the Index Pattern documents in the
.kibana index. Most of the time, saved objects in Kibana can be exported to a backup and then re-imported to restore the backup, but unfortunately, Index Pattern objects are not exported. There is an open issue in Github on this: https://github.com/elastic/kibana/issues/4288
The alternative to using exported saved objects is to create a snapshot that includes your
.kibana index. This is done using Elasticsearch APIs: https://www.elastic.co/guide/en/elasticsearch/reference/current/modules-snapshots.html
If you make a snapshot of your
.kibana index, then you can restore that into newly installed clusters, and then you'll be able to preserve all index patterns and their formatted fields.
Hope that helps!