What strategies exist to transition DB Migrations with Stored Procedures to Continuous Delivery?
It is quite difficult to migrate the databases without downtime, however, using some approaches you can do it with very minimum impact, nearly a zero downtime.

The main idea in this is to create new databases with new schema. Then make changes in your code. First migrate the data from the old database to the new database. Then make changes in your code such that it starts writing to both the databases, old and new. You need to decide a time (time when there will be least writes to the database). Then stop all the writes. Remove the old database and make sure all the writes should now happen to new database from now on. 
Then start the writes back. 

In this way, you can achieve DB migration with minimal downtime.

You can check this for a detailed information on how this can be done.