My usual path for any upgrade is as follows:
Shut down kibana app
Update our non-datanodes (coordinators, masters, read , etc) one at a time.
Update each data node, wait for a majority of the shards to recover before proceeding to the next.
Once everything is on a new elasticsearch version, remove a kibana instance from the load balancer and upgrade then start kibana so it is the only node trying to migrate the kibana index.
once it completes, add it back so service is restored, then upgrade/restart all the other kibana nodes and they will eventually come back online.
Caveats –
if you use s3 for snapshots you have to upgrade the s3 plugin across all nodes before elasticsearch is restarted.
kibana may need some configuration adjustments.
make sure you have a policy for removing the default certificates for cluster communication.
ymmv.