You need to instantiate a secondary DB replica somewhere else and start the DB migration. Since there will be "two instances" of the same DB running, you will also need to set up a (temporary) proxy for routing and handling the DB requests w/ something like this:
1) if the data being requested is already migrated, request is handled by the (new) secondary replica. 2) Primary instance handles the request, otherwise 2.1) Requested data should be migrated to secondary replica (asynchronously, but note that a repeated request may invalidate a migration).
Turn the proxy router off once the whole state of your primary DB instance is fully migrated, making the secondary replica the primary one. That's really just a napkin recipe for completing a live migration, though.
* We are now getting into the distributed transaction world because you can never be 100% sure that writing to 2 databases can succeed or fail at the same time. There is this talk from this guy who deals with similar problem you have: http://www.aviransplace.com/2015/12/15/safe-database-migrati...
I am not expert, but K8s handles task replication, and either spawn or route a request to another task instance somewhere else. However, the application logic itself must handle the fault-tolerance (by handling its states through transactions or something else) should an instance fail. K8s doesn't do that for you.